Math Problems Vault

Problem 10.1

(a)(b)

Let a random process $\xi_{t}(\omega )$ be given on the probability space $(\Omega , \mathscr {A}, P)$ , where $\Omega = \left\{ 1,2\right\} , \mathscr {A}$ is the set of all subsets of $\Omega$ , and $P$ assigns probability $1/2$ to each of the sets $\left\{ 1\right\}$ and $\left\{ 2\right\}$ . Let the set of values of the parameter $t$ be the segment $[0,1]$ , and $\xi_{t}(\omega ) = \omega t$ . Find:

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(a)

all realizations of the process $\xi_{t}(\omega )$ ;

(b)

all two-dimensional, three-dimensional, and $n$ -dimensional distributions of the process $\xi_{t}(\omega )$ .

Problem 10.2

(a)(b)

Let a random process $\xi_{t}(\omega )$ be defined on the probability space $(\Omega , \mathscr {A}, P), \Omega = [0,1], \mathscr {A}$ the $\sigma$ -algebra of Borel subsets, $P$ the Lebesgue measure, $t \in [0,1]$ , and

\xi _{t}(\omega ) = \begin{cases} 1, & \text{ for } t \leqslant \omega , \\ 0, & \text{ for } t > \omega . \end{cases}

Find:

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(a)

all realizations of the process $\xi_{t}(\omega )$ ;

(b)

the two-dimensional distributions of the process $\xi_{t}(\omega )$ .

Problem 10.3

Let $\eta$ be a random variable with distribution function $F(x)$ , $t \in R$ . Find all finite-dimensional distributions of the random process $\xi_{t} = \eta +t$ .

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Problem 10.4

Let $\eta$ and $\zeta$ be independent random variables having the same normal distribution with mean zero and variance $1/2$ , $t > 0$ . Find all finite-dimensional distributions of the random process $\xi_{t} = (\eta +\zeta )/t$ .

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Problem 10.5

Let $\xi$ and $\eta$ be random variables, where $\eta$ has a distribution symmetric about zero and $\mathbb {P}\left(\eta = 0\right) = 0$ . Find the probability that the realizations of the random process

\zeta _{t} = \xi +t(\eta +t), \quad t \geqslant 0,

are increasing.

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Problem 10.6

Let $\eta_{1}$ and $\eta_{2}$ be independent random variables having the same uniform distribution on the segment $[-1,1]$ , $t = \left(t_{1}, t_{2}\right) \in R^{2}$ . Find the values of $a$ for which almost all realizations of the random function $t_{1}\left(\eta_{1}+t_{2}\left(\eta_{2}+2 a\right)\right)$ are monotonically increasing in $t_{1}$ for $t_{2} = a$ .

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Problem 10.7

Give an example of a random process $\xi_{t}$ such that the set of elementary events to which continuous realizations of the process $\xi_{t}$ correspond is not itself an event.

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Problem 10.8

Prove that if a random process $\xi_{t}, t \in R$ , is stochastically continuous on a compact set $A \subset R$ , then it is uniformly stochastically continuous on this set.

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Problem 10.9

Prove that if a random process is stochastically continuous on a compact set $A \subset R$ , then it is stochastically bounded on this set.

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Problem 10.10

Let $\xi_{t}$ be a stochastically continuous process, and $g(x)$ a continuous function. Prove that the process $g\left(\xi_{t}\right)$ is also stochastically continuous.

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Problem 10.11

Give an example of a random process, stochastically continuous on a segment, all of whose trajectories are discontinuous.

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Problem 10.12

Let $\xi_{t}, t \in [0,1]$ , be a random process such that all $\xi_{t}$ are mutually independent and have the same nondegenerate distribution. Prove that this process is not stochastically continuous at any point.

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Problem 10.13

Let a random process $\xi_{t}$ be continuous in the mean of order $p > 0$ on a compact set $A$ . Prove that $\xi_{t}$ is uniformly continuous in the mean of order $p$ on the set $A$ .

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Problem 10.14

Let a random process $\xi_{t}$ be continuous in the mean of order $p > 0$ on a compact set $A$ . Prove that there exists a positive number $C < \infty$ such that

\mathbb {E}\left[\left|\xi _{t}\right|^{p}\right] \leqslant C \quad \text{ for } \quad t \in A .

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Problem 10.15

Prove that for a random process $\xi_{t}$ to be stochastically continuous on a set $T$ , it is necessary and sufficient that for any $t_{0}, s_{0} \in T$

\lim _{t \rightarrow t_{0}, s \rightarrow s_{0}} \mathbb {P}\left(\xi _{t} < x_{1}, \xi _{s} < x_{2}\right) = \mathbb {P}\left(\xi _{t_{0}} < x_{1}, \xi _{s_{0}} < x_{2}\right),

for all $x_{1}, x_{2}$ at which $\mathbb {P}\left(\xi_{t_{0}} < x_{1}, \xi_{s_{0}} < x_{2}\right)$ is continuous.

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Problem 10.16

Let $\xi_{t}, a \leqslant t \leqslant b$ , be a stochastically continuous process, $f(t)$ a nonrandom function defined on $[a, b]$ . Prove that the random process $\eta_{t} = \xi_{t}+f(t)$ is stochastically continuous at exactly those points of the segment $[a, b]$ at which the function $f(t)$ is continuous.

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Problem 10.17

Let $\xi_{t}(\omega ), 0 \leqslant t \leqslant 1$ , be a measurable random process defined on the probability space $(\Omega , \mathscr {A}, P)$ , and let $\tau (\omega )$ be a random variable defined on the same probability space, with $\mathbb {P}\left(0 \leqslant \tau \leqslant 1\right) = 1$ . Prove that $\xi_{\tau } = \xi_{\tau (\omega )}(\omega )$ is a random variable.

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Problem 10.18

(a)(b)(c)

A random process $\xi_{t},-\infty < t < \infty$ , and a random variable $\tau$ are defined on the same probability space. Is the function $\xi_{\tau }$ always a random variable if:

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(a)

$\tau$ takes finitely many values;

(b)

$\tau$ takes countably many values;

(c)

$\tau$ is an arbitrary random variable?

Problem 10.19

Let $\xi_{t}$ be a stochastically continuous random process. Prove that for every continuous bounded function $\varphi (x)$ the function $\mathbb {E}\left[\varphi \right]\left(\xi_{t}\right)$ is continuous in $t$ .

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Problem 10.20

Give an example of a random process $\xi_{t}, t \in [a, b]$ , such that for every continuous bounded function $g(x)$ the function $\mathbb {E}\left[g\right]\left(\xi_{t}\right)$ is continuous on $[a, b]$ , but $\xi_{t}$ is not stochastically continuous.

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Problem 10.21

Let $\xi_{t}$ be a stochastically continuous random process, and $g(x)$ a continuous function. Prove that if for some $\alpha > 1$

\sup _{t} \mathbb {E}\left[\left|g\left(\xi _{t}\right)\right|^{\alpha }\right] < \infty ,

then the function $\mathbb {E}\left[g\right]\left(\xi_{t}\right)$ is continuous in $t$ .

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Problem 10.22

Show that stochastically equivalent processes have identical finite-dimensional distributions.

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Problem 10.23

Prove that a process stochastically equivalent to a stochastically continuous process is stochastically continuous.

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Problem 10.24

Consider, on the probability space $(\Omega , \mathscr {A}, P)$ consisting of the segment $[0,1]$ with the $\sigma$ -algebra of Borel subsets and the Lebesgue measure, the random process $\xi_{t}(\omega )$ defined as follows:

\xi _{t}(\omega ) = \begin{cases} 1, & \text{ if the line through the point }(t, \omega ) \text{ parallel to the line } t = \omega \text{ crosses the } t\text{-axis at a rational point}, \\ 0, & \text{ otherwise. } \end{cases}

Show that $\xi_{t}(\omega )$ is stochastically continuous, but all of its trajectories are discontinuous at every point.

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Problem 10.25

Let $\xi_{t}$ be a random process defined on the probability space $(\Omega , \mathscr {A}, \mathbb {P})$ . Prove that if $\Omega$ is countable and all singletons have positive probability, then the stochastic continuity of the process $\xi_{t}$ is equivalent to the continuity of all of its trajectories.

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Problem 10.26

Prove that if the set of values of the parameter $t$ of a random process $\xi_{t}$ is countable, then the process is measurable.

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Problem 10.27

Let $\xi_{t}$ be a random process, all of whose trajectories are continuous, and let the set of values of the parameter $t$ be a line segment. Prove that $\xi_{t}$ is measurable.

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Problem 10.28

Prove that all trajectories of a measurable random process are measurable. Must a random process be measurable if all of its trajectories are measurable?

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Problem 10.29

Let $\xi_{t}$ , $a \leqslant t \leqslant b$ , be a random process. Prove that for its measurability it suffices that all of its trajectories be right-continuous (or left-continuous).

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Problem 10.30

Let $\xi_{t}, a \leqslant t \leqslant b$ , be a stochastically continuous process. Prove that on $[a, b]$ there exists a measurable process $\xi_{t}^{*}$ , stochastically equivalent to $\xi_{t}$ .

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Problem 10.31

Let $\xi_{t}$ be a random process stochastically continuous on $[a, b]$ (except at no more than countably many points of $[a, b]$ ). Prove that on $[a, b]$ there exists a measurable process $\xi_{t}^{*}$ , stochastically equivalent to $\xi_{t}$ .

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Problem 10.32

Let $\xi_{t}, t \in T$ , be a random process, and let $N = \left\{ t_{1}, \ldots , t_{n}, \ldots \right\}$ be a countable set, dense in $T$ . The process is called $N$ -separable if for every $t \in T$

\lim _{\delta \rightarrow 0} \sup _{\left|t-t_{n}\right| < \delta } \xi _{t_{n}} \geqslant \xi _{t} \geqslant \lim _{\delta \rightarrow 0} \inf _{\left|t_{n}-t\right| < \delta } \xi _{t_{n}}

Let $T = [a, b]$ . Prove that if $\xi_{t}$ is stochastically continuous on $[a, b]$ , then for every countable set $N$ , dense in $[a, b]$ , there exists a process $\xi_{t}^{\prime }$ , stochastically equivalent to $\xi_{t}$ , that is $N$ -separable.

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Problem 10.33

Suppose all trajectories of a Poisson process are right-continuous. Prove that then almost all trajectories are nondecreasing integer-valued functions, increasing only by jumps of size 1.

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Problem 10.34

Prove that for a random process $\xi_{t}$ to be continuous in the mean square on a set $T$ , it is necessary and sufficient that the function $L(t, s) = \mathbb {E}\left[\xi_{t} \bar{\xi }_{s}\right]$ be jointly continuous in its arguments on the set $T \times T$ .

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Problem 10.35

Prove that for a random process $\xi_{t}$ to be continuously differentiable in the mean square on the interval $(a, b)$ , it is necessary and sufficient that the function $L(t, s) = \mathbb {E}\left[\xi_{t} \bar{\xi }_{s}\right]$ have a continuous mixed second-order partial derivative in $t$ and $s$ on the set $(a, b) \times (a, b)$ .

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Problem 10.36

Prove that a Poisson process is differentiable in probability, but not differentiable in the sense of convergence in the mean of any order $p \geqslant 1$ .

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Problem 10.37

Find the correlation function of the random process $\xi_{t} = \gamma_{1} f_{1}(t)+\ldots +\gamma_{n} f_{n}(t)$ , where $f_{1}(t), \ldots , f_{n}(t)$ are nonrandom functions, and $\gamma_{1}, \ldots , \gamma_{n}$ are uncorrelated random variables with variances $d_{1}, \ldots , d_{n}$ , respectively.

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Problem 10.38

Let $\xi_{t}, t \in T$ , be a random process with correlation function $K(t, s)$ . Prove that $K(t, s)$ is nonnegative definite.

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Problem 10.39

Let $\xi_{t}^{(1)}, \ldots , \xi_{t}^{(n)}$ be independent random processes such that $\mathbb {E}\left[\xi_{t}^{(i)}\right] = 0, i = 1, \ldots , n$ ; let $K_{1}(t, s), \ldots , K_{n}(t, s)$ be the corresponding correlation functions. Find the correlation function of the process $\xi_{t}^{(1)}+\ldots +\xi_{t}^{(n)}$ .

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Problem 10.40

Prove the positive definiteness of the following functions:

\begin{aligned} & K_{1}(t, s) = \min [t, s], \quad t, s \geqslant 0 \\ & K_{2}(t, s) = \begin{cases} 1-\left|t-s\right|, & \left|t-s\right| < 1 \\ 0, & \left|t-s\right| \geqslant 1, \quad t, s \in R \end{cases} \\ & K_{3}(t, s) = \min [t, s]-t s, \quad t, s \in [0,1] \\ & K_{4}(t, s) = e^{-\left|t-s\right|}, \quad t, s \in R \end{aligned}

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Problem 10.41

(a)(b)

Let $K(t, s), t, s \in T$ , be the correlation function of some random process, and $Q(z)$ a polynomial with positive coefficients. Prove that the function $K_{1}(t, s) = Q(K(t, s))$ is also the correlation function of some random process.

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(a)

(b)

Problem 10.42

Find the spectral density of the random process $\xi_{t}$ whose correlation function equals $K(t) = c e^{-\alpha \left|t\right|}, c, \alpha > 0$ , $t \in R$ .

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Problem 10.43

Let $\xi_{t}, t \in R$ , be a random process with mean zero and correlation function $K(t, s) = e^{s t}$ . Prove that it is infinitely differentiable in the mean square.

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Problem 10.44

Let $\xi$ be a random variable having a normal distribution with mean $m$ and variance $\sigma^{2}$ , $b$ a real number. Find the correlation function of the process

\xi _{t} = \xi t+b, t \geqslant 0

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Problem 10.45

Let $A, \eta$ and $\varphi$ be random variables, with $\varphi$ independent of $A$ and $\eta$ , $A \geqslant 0, \eta \geqslant 0$ , and $\varphi$ uniformly distributed on the segment $[0,2 \pi ]$ . Find the mean and correlation function of the process

\xi _{t} = A \cos (\eta t+\varphi ), \quad t \in R .

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Problem 10.46

Let $\varphi_{1}(t), \ldots , \varphi_{n}(t)$ be arbitrary real functions, $c_{1}, \ldots , c_{n}$ nonnegative numbers. Prove that the function

K\left(t_{1}, t_{2}\right) = \sum _{i = 1}^{n} c_{i} \varphi _{i}\left(t_{1}\right) \varphi _{i}\left(t_{2}\right)

is the correlation function of some random process.

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Problem 10.47

Let $\xi_{t}^{(1)}$ and $\xi_{t}^{(2)}$ be two independent random processes with correlation functions $K_{1}(t, s)$ and $K_{2}(t, s)$ respectively. Find the correlation function of the process

\eta _{t} = \xi _{t}^{(1)} \xi _{t}^{(2)} .

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Problem 10.48

(a)(b)(c)

Find the generating function of the number of particles in the $n$ -th generation, if the generating function of the immediate descendants of a single particle equals:

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(a)

$p z+1-p$ ;

(b)

$(1-p) /(1-p z)$ ;

(c)

$1-p(1-z)^{\alpha }, \quad 0 < p < 1$ , $0 < \alpha < 1$ .

Problem 10.49

(a)(b)(c)

Find the extinction probabilities for branching processes with generating function of the number of descendants of a single particle:

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(a)

$(1-p) /(1-p z)$ ;

(b)

$1-p(1-z)^{\alpha }, 0 < \alpha < 1$ ;

(c)

$\left(1+z+z^{2} + z^{3}\right) / 4$ .

Problem 10.50

(a)(b)

Find the distribution of the extinction time $T$ for branching processes with generating function of the number of descendants of a single particle:

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(a)

$p z+1-p$ ;

(b)

$1-p(1-z)^{\alpha }, 0 < \alpha < 1$ .

Problem 10.51

Let $X_{0}, X_{1}, \ldots$ be a branching process, $X_{0} = 1$ . Prove that

\mathbb {P}\left( X_{n} > N \text{ for some } 1 \leqslant n \leqslant m-1 \mid X_{m} = 0 \right) \leqslant \left[\mathbb {P}\left(X_{m} = 0\right)\right]^{N} .

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Problem 10.52

Find the generating function of the total number of particles in the first $n$ generations, if the generating function of the number of descendants of a single particle equals $p z+1-p$ .

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Problem 10.53

Consider the branching process $X_{0}, X_{1}, \ldots$ with generating function

\begin{gathered} \varphi (z) = \frac{1-(b+c)}{1-c}+\frac{b z}{1-c z} \\ 0 < c < b+c < 1, \quad (1-b-c) /[c(1-c)] > 1. \end{gathered}

Find $\lim_{n \rightarrow \infty } \mathbb {P}\left( X_{n} = k \mid X_{n} > 0 \right)$ .

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Problem 10.54

Let, in the branching process $X_{0}, X_{1}, \ldots$ of problem 10.53, $1-b-c = c(1-c)$ . Find $\lim_{n \rightarrow \infty } \mathbb {P}\left( X_{n} \leqslant n x \mid X_{n} > 0 \right)$ .

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Problem 10.55

Let $X_{0}, X_{1}, \ldots$ be a branching process, $X_{0} = 1, \mathbb {E}\left[X_{1}\right] = m$ . Prove that

\mathbb {E}\left[ X_{n+k} \mid X_{n} \right] = m^{k} X_{n} .

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Problem 10.56

Let $X_{0}, X_{1}, \ldots$ be a branching process, $X_{0} = 1, \varphi (z) = \mathbb {E}\left[z^{X_{1}}\right], \quad \varphi_{n}(z) = \mathbb {E}\left[z^{X_{n}}\right], \quad \varphi^{\prime }(1) > 1$ . Denote by $Y_{n}$ the number of all particles in the $n$ -th generation that have infinitely many generations of descendants. Prove that

\sum _{k = 0}^{\infty } z^{k} \mathbb {P}\left( Y_{n} = k \mid Y_{0} = X_{0} = 1 \right) = \frac{\varphi _{n}(z(1-q)+q)-q}{1-q} .

where $q$ is the extinction probability.

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Problem 10.57

(a)(b)(c)(d)

Find the generating function of the number of particles at time $t$ for a branching process with continuous time and generating function of infinitesimal parameters:

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(a)

$f(z) = a_{2} z^{2}+a_{1} z-\left(a_{1}+a_{2}\right), a_{2} > 0$ ;

(b)

$f(z) = z^{k}-z, k \geqslant 2$ ;

(c)

$f(z) = 1-z-\sqrt{1-z}$ ;

(d)

$f(z) = \lambda (1-z)[1+\ln (1-z)], \lambda > 0$ .

Problem 10.58

(a)(b)(c)(d)

Find the extinction probabilities of the branching processes from problem 10.57.

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(a)

(b)

(c)

(d)

Problem 10.59

Let $X_{t}$ be a branching process with generating function of infinitesimal parameters $f(z) = a_{2} z^{2}+a_{1} z-\left(a_{1}+a_{2}\right)$ , where $a_{2} > 0, a_{1}+2 a_{2} < 0$ . Prove that

\lim _{t \rightarrow \infty } e^{-\left(a_{1}+2 a_{2}\right) t} \mathbb {P}\left(X_{t} > 0\right) = 1+\frac{a_{2}}{a_{1}+a_{2}} .

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Problem 10.60

Prove that for the branching process $X_{t}$ with generating function $f(z) = 3-5 z+z^{2}+z^{3}$

\lim _{t \rightarrow \infty } t^{-1} \mathbb {P}\left(X_{t} > 0\right) = \frac{1}{4} .

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Problem 10.61

Let $X_{t}$ be a branching process with generating function $f(z) = z^{2}-z$ . Prove that $X_{t} e^{-t}$ converges as $t \rightarrow \infty$ in the mean square to a random variable $\xi$ having an exponential distribution with parameter 1.

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Problem 10.62

Let $X_{t}$ be a branching process with generating function $f(z) = z^{2}-2 z+1$ . Prove that

\lim _{t \rightarrow \infty } \mathbb {P}\left( \frac{X_{t}}{\mathbb {E}\left[X_{t} \mid X_{t} > 0\right]} < x \mid X_{t} > 0 \right) = 1-e^{-x} .

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Problem 10.63

Prove that the following definitions of a Markov process $\xi_{t}$ are equivalent:

for every $t$ and any $A \in \mathscr {F}_{ \leqslant t}, B \in \mathscr {F}_{ \geqslant t}$
$\mathbb {P}\left( A B \mid \mathscr {F}_{ = t} \right) = \mathbb {P}\left( A \mid \mathscr {F}_{ = t} \right) \mathbb {P}\left( B \mid \mathscr {F}_{ = t} \right) \text{ a.s.; }$
for every $t$ and any $B \in \mathscr {F}_{\geqslant t}$ :
$\mathbb {P}\left( B \mid \mathscr {F}_{ \leqslant t} \right) = \mathbb {P}\left( B \mid \mathscr {F}_{ = t} \right) \text{ a.s.; }$
for every $t$ and any $A \in \mathscr {F}_{\leqslant t}$
$\mathbb {P}\left( A \mid \mathscr {F}_{ > t} \right) = \mathbb {P}\left( A \mid \mathscr {F}_{ = t} \right) \text{ a.s. }$

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Problem 10.64

Prove that a random process $\xi_{t}, t \in T$ , with values in the phase space $(X, \mathscr {B})$ is Markov if and only if for any $s_{1} \leqslant s_{2} \leqslant \ldots \leqslant s_{m} \leqslant t \leqslant t_{1} \leqslant \ldots \leqslant t_{n}$ , $s_{i}, t, t_{j} \in T$ , and any $A_{1}, \ldots , A_{m}, B_{1}, \ldots , B_{n} \in \mathscr {B}$

\begin{aligned} \mathbb {P}\left( \bigcap _{ i = 1 } ^{ m } \left\{ \xi _{ s _{ i } } \in A _{ i } \right\} \cap \bigcap _{ j = 1 } ^{ n } \left\{ \xi _{t_{j}} \in B_{j}\right\} \mid \xi _{t} \right) = \\ & = \mathbb {P}\left( \bigcap _{i = 1}^{m}\left\{ \xi _{s_{i}} \in A_{i}\right\} \mid \xi _{t} \right) \mathbb {P}\left( \bigcap _{j = 1}^{n}\left\{ \xi _{t_{j}} \in B_{j}\right\} \mid \xi _{t} \right) . \end{aligned}

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Problem 10.65

For a random process $\xi_{t}, t \in T$ , with values in the phase space $(X, \mathscr {B})$ to be Markov, it is necessary and sufficient that for any $s_{1} \leqslant s_{2} \leqslant \ldots \leqslant s_{m} \leqslant t \leqslant t_{1} \leqslant \ldots \leqslant t_{n}$ , $s_{i}, t, t_{j} \in T$ , and any bounded $\mathscr {B}$ -measurable functions $f_{1}, \ldots , f_{m}, g_{1}, \ldots , g_{n}$ , $\mathbb {E}\left[ \prod_{i = 1}^{m} f_{i}\left(\xi_{s_{i}}\right) \prod_{j = 1}^{n} g_{j}\left(\xi_{t_{j}}\right) \mid \xi_{t} \right] =\mathbb {E}\left[ \prod_{i = 1}^{m} f_{i}\left(\xi_{s_{i}}\right) \mid \xi_{t} \right] \mathbb {E}\left[ \prod_{j = 1}^{n} g_{j}\left(\xi_{t_{j}}\right) \mid \xi_{t} \right]$ a.s.

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Problem 10.66

Let $\xi_{t}, t \geqslant 0$ , be a Markov process. Prove that $\xi_{n}$ , $n = 0,1, \ldots$ , is a Markov process with discrete time.

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Problem 10.67

Let $\xi_{t}, t \geqslant 0$ , be a Markov process. Is the sequence

\eta _{n} = \left[\xi _{n}\right],

where $[x]$ is the integer part of $x$ , a Markov chain?

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Problem 10.68

(a)(b)(c)

Let $\left\{ \xi_{n}\right\}$ be a sequence of independent random variables having the same density $p(x)$ , $p(x) > 0,-\infty < x < \infty$ . Is the sequence $\left\{ \eta_{n}\right\}$ Markov if:

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(a)

$\eta_{n} = \xi_{n}, n = 0,1, \ldots$ ;

(b)

$\eta_{n} = \xi_{0}+\ldots +\xi_{n}, n = 0,1, \ldots$ ;

(c)

$\eta_{n} = \max \left\{ 0, \xi_{0}, \ldots , \xi_{n}\right\}$ .

For the Markov chains, find the one-step transition probabilities.

Problem 10.69

Let $\left\{ \xi_{n}\right\}$ be a sequence of independent random variables having the same density $p(x)$ , $p(x) > 0,-\infty < x < \infty$ . Set

\begin{aligned} & S_{t} = S_{k}(t-k)+S_{k+1}(k+1-t), \quad k \leqslant t \leqslant k+1, \\ & S_{n} = \xi _{1}+\ldots +\xi _{n} . \end{aligned}

Is the process $S_{t}$ Markov?

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Problem 10.70

Let $\xi_{t}, t \geqslant 0$ , be a homogeneous Markov process with countably many states $(0,1, \ldots )$ . Prove that if the transition functions $P_{ij}(t)$ are continuous at $t = 0$ , then they are uniformly continuous for $t \geqslant 0$ .

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Problem 10.71

Let $\xi_{t}$ and $\eta_{t}, t \geqslant 0$ , be two homogeneous Markov processes with countably many states, $P_{ij}(t)$ and $Q_{ij}(t)$ the corresponding transition functions. Prove that if for some $t_{0} > 0$ and all $i$ and $j$ , $P_{ij}(t) = Q_{ij}(t)$ for $0 \leqslant t \leqslant t_{0}$ , then $P_{ij}(t) = Q_{ij}(t)$ for all $t \geqslant 0$ .

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Problem 10.72

Let $\xi_{t}, t \geqslant 0$ , be a homogeneous Markov process with finitely many states $\left\{ 0,1, \ldots , N\right\}$ and transition functions $P_{ij}(t)$ . Prove that the determinant of the matrix $P(t)$ with entries $P_{ij}(t), 0 \leqslant i, j \leqslant N$ , is positive for all $t > 0$ .

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Problem 10.73

Let $\xi_{t}, t \geqslant 0$ , be a homogeneous Markov process with finitely many states $\left\{ 0,1, \ldots , N\right\}$ and transition functions $P_{ij}(t)$ . Suppose that $P_{ij}(t)$ is continuous for all $t \geqslant 0$ . Prove that the finite limits

a_{i j} = \lim _{h \rightarrow 0} \frac{P_{ij}(h)-\delta _{ij}}{h}, \quad \delta _{ij} = \begin{cases} 1, & i \neq j, \\ 0, & i \neq j, \end{cases}

exist, and $\sum_{j \neq i} a_{ij} = -a_{ii}$ .

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Problem 10.74

Give an example of a homogeneous Markov process with countably many states for which the finite limits

\lim _{t \rightarrow 0} \frac{P_{ii}(t)-1}{t} = a_{ii}, \quad \lim _{t \rightarrow 0} \frac{P_{ij}(t)}{t} = a_{ij}, \quad i \neq j

exist, but $\sum_{j \neq i} a_{ij} \neq -a_{ii}$ .

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Problem 10.75

Prove that any homogeneous Markov chain is strongly Markov with respect to the family of $\sigma$ -algebras $\mathscr {F}_{\leqslant n}$ , $n = 0,1, \ldots$ .

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Problem 10.76

Give an example of a Markov, but not strongly Markov, family.

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Problem 10.77

Let $\xi_{t}, t \geqslant 0$ , be a Markov random process with countable state space $\left\{ 0,1, \ldots \right\}$ . Suppose that at time $t = 0$ the process is in state $i$ . Find the distribution function of the time until the first change of state of the process.

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Problem 10.78

Find all invariant measures corresponding to the transition probability matrix

P(t) = \left(P_{ij}(t)\right) =\left[\begin{smallmatrix} \left(2+e^{-3 t}\right) / 3 & \left(1-e^{-3 t}\right) / 3 \\ \left(2-2 e^{-3 t}\right) / 3 & \left(1+2 e^{-3 t}\right) / 3 \end{smallmatrix}\right]

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Problem 10.79

Give an example of a Markov family with two finite invariant measures that are not proportional to one another.

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Problem 10.80

Show that there is no Markov process $\xi_{t}$ , $t \geqslant 0, \xi_{0} = 0$ , with two states $\left\{ 0,1\right\}$ and trajectories almost surely continuous, whose transition probabilities equal $P_{00}(t) = e^{-t}, P_{01}(t) = 1-e^{-t}, P_{11}(t) = 1$ .

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Problem 10.81

Let $\xi$ and $\eta$ be independent nonnegative random variables with

\mathbb {P}\left(\xi < x\right) = 1-e^{-a x}, \quad \mathbb {P}\left(\eta < x\right) = G(x) .

Prove that

\mathbb {P}\left( \xi < u+\eta \mid \xi \geqslant \eta \right) = 1-e^{-a u}, u \geqslant 0,

in particular, for every $t \geqslant 0$ ,

\mathbb {P}\left( \xi < u+t \mid \xi \geqslant t \right) = 1-e^{-a u}, u \geqslant 0.

(The memoryless property of the exponential distribution.)

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Problem 10.82

Prove that a random flow is Poisson with intensity $\lambda$ if and only if it is a recurrent flow with $A(t) = 1-e^{-\lambda t}, t \geqslant 0$ .

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Problem 10.83

Prove that the random flow $v_{t}$ , obtained by superposition of $k$ independent Poisson flows $v_{t}^{(1)}, \ldots , v_{t}^{(k)}$ with intensities $\lambda_{1}, \ldots , \lambda_{k}$ , is Poisson with intensity $\lambda = \lambda_{1}+\ldots +\lambda_{k}$ .

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Problem 10.84

Let a Poisson flow with intensity $\lambda$ be given. Assign each request of this flow, with probability $p_{i}, i = 1, \ldots , k$ , $\sum_{i = 1}^{k} p_{i} = 1$ , to the $i$ -th subflow, independently of the other requests. Prove that the $i$ -th subflow is Poisson with intensity $\lambda p_{i}$ .

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Problem 10.85

Let a Poisson flow with intensity $\lambda$ be subjected to the following thinning operation: the first $k$ requests are lost, the $(k+1)$ -th is kept, then again $k$ are lost, the next is kept, and so on. Prove that the thinned flow (called an Erlang flow of order $k$ ) is a recurrent flow determined by the distribution function

A(t) = \int _{0}^{\lambda t} \frac{x^{k}}{k!} e^{-x} d x

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Problem 10.86

Denote by $v_{t}$ the random number of requests of a Poisson flow that arrive in the interval $[0, t)$ . Prove that for $t \leqslant T, k \leqslant n$ ,

\mathbb {P}\left( v_{t} = k \mid v_{T} = n \right) = C_{n}^{k}\left(\frac{t}{T}\right)^{k}\left(1-\frac{t}{T}\right)^{n-k} .

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Problem 10.87

Prove that if it is known that $N$ requests of a Poisson flow arrived on the segment $[0, T], T > 0$ , then the flow of requests on this segment is a Bernoulli flow.

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Problem 10.88

Let a recurrent flow of requests be given, determined by the distribution function $A(t)$ . Each request, independently of the others, is either discarded from the flow with probability $p$ , or kept with probability $1-p$ . Show that the flow of kept requests is a recurrent flow determined by the distribution function $B(t)$ , where

B(t) = A(t)-p \int _{0}^{t} P(p, t-u) d A(u),

and the Laplace transform of the function $P(z, t)$ equals

\int _{0}^{\infty } e^{-s t} P(z, t) d t = \frac{1}{s} \frac{1-\alpha (s)}{1-z \alpha (s)} \quad \alpha (s) = \int _{0}^{\infty } e^{-s t} d A(t)

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Problem 10.89

Prove that for any stationary flow there exists

\lim _{t \rightarrow 0} \frac{1-P_{0}(t)}{t} = \mu ,

where $P_{0}(t)$ is the probability that no request arrives in time $t$ . The number $\mu$ (finite or infinite) is called the parameter of the stationary flow.

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Problem 10.90

Let $\mu$ be the parameter, and $\lambda$ the intensity (i.e. the mean number of requests arriving per unit time) of a stationary flow. Prove that if the parameter is a finite positive number, then the flow is ordinary $\iff \lambda = \mu$ .

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Problem 10.91

For each $n \geqslant 1$ consider the total flow $\sum_{n}$ , obtained by superposition of $n$ independent flows, where the $k$ -th flow ( $k = 1, \ldots , n$ ) is a delayed recurrent flow determined by the distribution functions $A_{1k}(t)$ and $A_{k}(t)$ :

A_{1 k}(t) = a_{k} \int _{0}^{t}\left[1-A_{k}(u)\right] d u, \quad a_{k}^{-1} = \int _{0}^{\infty }\left[1-A_{k}(u)\right] d u .

Suppose that as $n \rightarrow \infty$ : 1) $a_{1}+\ldots +a_{n} = a = \text{const}$ ; 2) $\max_{1 \leqslant k \leqslant n} \left\{ a_{k}\right\} \rightarrow 0$ ; 3) for each fixed $t$ ,

\max _{1 \leqslant k \leqslant n}\left\{ A_{k}(t)\right\} \rightarrow 0.

Prove that as $n \rightarrow \infty$ the flow $\sum_{n}$ converges uniformly to a Poisson flow with parameter $a$ .

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Problem 10.92

The flow of passengers to a bus stop is Poisson with intensity $\lambda$ . At random time intervals $\xi_{1}, \xi_{2}, \ldots$ , buses arrive at the stop. The random variables $\xi_{1}, \xi_{2}, \ldots$ are mutually independent and identically distributed with a nonlattice distribution function $G(x)$ , $\quad \mu^{-1} = \int_{0}^{\infty } x d G(x) < \infty$ . A bus picks up all passengers present at the stop at the moment of its arrival. Let $w_{t}$ be the waiting time until the next bus arrives, starting from time $t$ . Find $\lim_{t \rightarrow \infty } \mathbb {P}\left(w_{t} < y\right)$ .

Problems 10.93--10.95 consider the same service system as in problem 10.92.

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Problem 10.93

Let $\alpha_{t}$ be the time elapsed since the most recent bus arrival up to time $t$ . Find $\lim_{t \rightarrow \infty } \mathbb {P}\left(\alpha_{t} < y\right)$ .

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Problem 10.94

Find $\lim_{t \rightarrow \infty } \mathbb {P}\left(w_{t} < u, \alpha_{t} < v\right)$ .

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Problem 10.95

Let $v_{t}$ be the number of passengers at the stop at time $t$ . Find $\lim_{t \rightarrow \infty } \mathbb {P}\left(v_{t} = k\right), \quad k = 0,1,2, \ldots$

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Problem 10.96

(a)(b)

Find the probability $P_{0}(t)$ that the system $M \mid M \mid 1 \mid 0$ is free at time $t$ , if the intensity of the input flow equals $\lambda$ , the mean service time is $\mu^{-1}$ , and:

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(a)

at time $t = 0$ the system was free;

(b)

at time $t = 0$ the system was busy.

Problem 10.97

Find the probability $q_{n}$ that, in the stationary regime of the system $M \mid M \mid n \mid \infty$ , all servers are busy, if the intensity of the input flow equals $\lambda$ , the mean service time is $\mu^{-1}$ , with $\lambda < n \mu$ .

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Problem 10.98

Consider the queueing system $M \mid M \mid 1 \mid \infty$ . Suppose in addition that the length of stay of the $n$ -th request (requests are numbered in order of their arrival in the system) in the queue is bounded by a random variable $\xi_{n}$ . The random variables $\xi_{1}, \xi_{2}, \ldots$ are mutually independent and identically distributed with distribution function $1-e^{-v x}, x \geqslant 0$ . Let $\lambda$ be the intensity of the input flow, $\mu^{-1}$ the mean service time. Find the probability $P_{0}$ that in the stationary regime the system is free.

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Problem 10.99

Consider the same queueing system as in the previous problem. Find the expectation $\bar{n}$ of the number of requests in the system in the stationary regime.

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Problem 10.100

(a)(b)

Find the stationary probability that in the system $M \mid M \mid n \mid m$ :

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(a)

all servers are busy ( $P_{n}$ );

(b)

all waiting places are occupied ( $P_{n+m}$ ).

Problem 10.101

Consider the same queueing system as in the previous problem. Find the expectation $\bar{N}$ of the number of servers busy with service in the stationary regime.

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Problem 10.102

(a)(b)

Consider the same queueing system as in the previous problem. Find the expectation of the number of requests:

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(a)

in the queue $\bar{q}$ ;

(b)

in the system $\bar{n}$ , in the stationary regime.

Problem 10.103

Consider the system $M \mid GI \mid 1 \mid \infty$ . Let $\lambda$ be the intensity of the input flow, $B(x)$ the distribution function of the service duration, $\pi_{j}$ the probability of finding $j$ requests in the system in the stationary regime. Prove that for

\begin{gathered} B(x) = \begin{cases} 0, & x \leqslant \alpha , \\ 1, & x > \alpha ,\end{cases} \\ \pi _{0} = 1-\lambda \alpha , \pi _{1} = \left(e^{\lambda \alpha }-1\right)(1-\lambda \alpha ), \\ \pi _{j} = (1-\lambda \alpha ) \sum _{k = 1}^{j-1}(-1)^{j-k} e^{k \lambda \alpha }\left[\frac{(k \lambda \alpha )^{j-k}}{(j-k)!}+\frac{(k \lambda \alpha )^{j-k-1}}{(j-k-1)!}\right]+ \\ \quad +(1-\lambda \alpha ) e^{j \lambda \alpha }, \quad j \geqslant 2. \end{gathered}

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Problem 10.104

Consider the same queueing system as in the previous problem. Let $f_{j}$ be the probability that $j$ requests are served during a busy period. Prove that for

B(x) = \begin{cases} 0, & x \leqslant \alpha , \\ 1, & x > \alpha , \end{cases} \quad f_{j} = \frac{e^{-\lambda \alpha j}}{j!}(\lambda \alpha j)^{j-1} .

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Problem 10.105

Consider the system $M \mid GI \mid 1 \mid \infty$ . Let $W(x)$ be the distribution function of the waiting time in the stationary regime, $\lambda$ the intensity of the input flow, $B(x)$ the distribution function of the service time. Prove that $W(x)$ is an infinitely divisible distribution function.

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Problem 10.106

Consider the same queueing system as in the previous problem. Find the distribution function of the time intervals between departures of requests from the system in the stationary regime.

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Problem 10.107

Consider the system $M \mid GI \mid 1 \mid \infty$ . Suppose that after the service of each request finishes, with probability $p$ it leaves the system, and with probability $1-p$ it returns to the queue for repeat service, independently of the other requests and of the number of previous services this request has received. Find the Laplace--Stieltjes transform $\pi (s)$ of the distribution function of the duration of a busy period.

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Problem 10.108

Consider the same queueing system as in the previous problem. Find the generating function $P(z)$ of the number of requests in the system in the stationary regime.

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Problem 10.109

Consider the same queueing system as in the previous problem. Find the distribution function $F(x)$ of the time intervals between requests departing the system in the stationary regime.

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Problem 10.110

Find the Laplace transform of the probability that the system $M \mid GI \mid 1 \mid 0$ is free at time $t$ , given that at time $t = 0$ the system was free.

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Problem 10.111

Find $\pi (s)$ , the Laplace--Stieltjes transform of the distribution function of the duration of a busy period in the system $M \mid GI \mid 1 \mid 1$ .

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Problem 10.112

Consider the same queueing system as in the previous problem. Find the generating function $f(z)$ of the number of requests served during a busy period.

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Problem 10.113

Consider the queueing system $M \mid GI \mid \infty$ . Let $\lambda$ be the intensity of the input flow, $B(x)$ the distribution function of the service time at any server, $n_{t}$ the number of requests in the system at time $t$ . Find the joint distribution of $(n_{t_{1}}, n_{t_{2}})$ , $t_{1} < t_{2}$ .

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Problem 10.114

Consider the same queueing system as in the previous problem. Let $\mu_{t}$ be the number of requests served up to time $t$ . Find the joint distribution of $(\mu_{t_{1}}, \mu_{t_{2}})$ , $t_{1} < t_{2}$ .

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Problem 10.115

Consider the same queueing system as in the previous problem. Find the joint distribution of $(n_{t}, \mu_{t})$ .

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Problem 10.116

Consider the queueing system $M \mid GI \mid 1 \mid \infty$ . Suppose in addition that the service duration of a request arriving to a free system has distribution function $B_{1}(t)$ , different from the distribution function $B(t)$ of the service duration of requests arriving to a busy system. Find the Laplace--Stieltjes transform $\pi (s)$ of the duration of a busy period.

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Problem 10.117

Consider the same queueing system as in the previous problem. Find the generating function $f(z)$ of the number of requests served during a busy period.

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Problem 10.118

Consider the queueing system $M \mid GI \mid 1 \mid \infty$ . Suppose that the server is unreliable while busy. The operating time of the server until breakdown has an exponential distribution $1-e^{-v x}, x \geqslant 0$ . Immediately after a breakdown, repair of the server begins, lasting a random time with distribution function $G(x)$ . The request during whose service the server broke down is lost. Let $\lambda$ be the intensity of the input flow, $B(x)$ the distribution function of the service time. Find the Laplace--Stieltjes transform of the duration of a busy period.

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Problem 10.119

Consider the same queueing system as in the previous problem. Find the Laplace--Stieltjes transform of the distribution function of the waiting time in the stationary regime.

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Problem 10.120

Consider the same queueing system as in the previous problem. Find the generating function $P(z)$ of the number of requests in the system in the stationary regime.

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Problem 10.121

Find the correlation function of the Wiener process $w_{t}$ .

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Problem 10.122

Let $w_{t}$ be a Wiener process. Find the joint density of the values $w_{u}$ and $w_{v}$ , $0 < u < v < 1$ , given that $w_{1} = 0$ .

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Problem 10.123

Let $w_{t}$ be a Wiener process. Find the covariance of the values $w_{s}$ and $w_{t}, s < t < 1$ , given that $w_{1} = 0$ .

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Problem 10.124

Let $w_{t}$ be a Wiener process. Find the correlation function of the process $w_{t}^{(0)} = w_{t}-t w_{1}$ , considered on the segment $0 \leqslant t \leqslant 1$ (the conditional Wiener process).

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Problem 10.125

Let $w_{t}^{(0)}, \quad 0 \leqslant t \leqslant 1$ , be the conditional Wiener process defined in the previous problem. Prove that the process $\bar{w}_{t} = (1 + t) w_{1 /(1+t)}^{(0)}, \quad t \geqslant 0,$ is a Wiener process.

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Problem 10.126

(a)(b)

Let $w_{t}$ be a Wiener process. Prove that the following processes are also Wiener processes:

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(a)

$w_{t}^{(1)} = \begin{cases} 0, & t = 0, \\ t w_{1 / t}, & t > 0 ; \end{cases}$

(b)

$\begin{aligned} w_{t}^{(2)} & = \sqrt{c} w_{t / c}, \quad t \geqslant 0, \\ c & = \text{const} > 0.\end{aligned}$

Problem 10.127

Let $w_{t}^{(1)}$ and $w_{t}^{(2)}$ be independent Wiener processes. Prove that the process $\frac{1}{\sqrt{2}}\left(w_{t}^{(1)}+w_{t}^{(2)}\right), \quad t \geqslant 0$ , is also a Wiener process.

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Problem 10.128

Let $w_{t}, t \geqslant 0$ , be a Wiener process. Set

w_{t}^{(0)} = \begin{cases} w_{t}, & t \leqslant T, \\ 2 w_{T}-w_{t}, & t > T . \end{cases}

Prove that $w_{t}^{(0)}$ is a Wiener process.

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Problem 10.129

Let $\xi_{0}, \xi_{1}, \xi_{2}, \ldots$ be independent random variables having the same normal distribution with mean zero and unit variance. Prove that

w_{t} = \frac{t}{\sqrt{\pi }} \xi _{0}+\sqrt{\frac{2}{\pi }} \sum _{n = 1}^{\infty } \sum _{k = 2}^{2^{n}-1} \frac{\sin k t}{k} \xi _{k}, \quad t \in [0, \pi ],

is a Wiener process.

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Problem 10.130

Prove that a Wiener process is not differentiable in probability.

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Problem 10.131

Let $w_{t}$ be a Wiener process. Prove that

\begin{aligned} \mathbb {E}\left[\left(w_{t}-w_{s}\right)^{2 n+1}\right] & = 0 \\ \mathbb {E}\left[\left(w_{t}-w_{s}\right)^{2 n}\right] & = (2 n-1)!!(t-s)^{n} \end{aligned}

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Problem 10.132

Prove that a Wiener process is Markov. Find its transition function.

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Problem 10.133

Find the finite-dimensional distributions of a Wiener process.

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Problem 10.134

Prove that almost all trajectories of a Wiener process are nowhere differentiable.

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Problem 10.135

Let $w_{t}$ be a Wiener process. Find the conditional density of $w_{t}, t_{1} < t < t_{2}$ , given that

w_{t_{1}} = A, \quad w_{t_{2}} = B .

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Problem 10.136

Let $w_{t}$ be a Wiener process. Prove that for $z \geqslant 0$

\mathbb {P}\left(\max _{0 \leqslant s \leqslant t} w_{s} \geqslant z\right) = 2 \mathbb {P}\left(w_{t} \geqslant z\right) .

Find the density of the random variable $\max_{0 \leqslant s \leqslant t} w_{s}$ .

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Problem 10.137

Let $\tau (z), z > 0$ , be the random time at which the Wiener process $w_{t}$ first reaches the value $z$ . Find the density of $\tau (z)$ . Show that the expectation of $\tau (z)$ is infinite.

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Problem 10.138

Let $\tau (z)$ be the random variable defined in the previous problem. Prove that the convolution of the distributions of the random variables $\tau \left(z_{1}\right)$ and $\tau \left(z_{2}\right)$ coincides with the distribution of the random variable $\tau \left(z_{1}+z_{2}\right)$ .

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Problem 10.139

Show that the distribution of the random variable $\tau (z)$ defined in problem 10.137 coincides with the distribution of the random variable $z^{2} \tau (1)$ .

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Problem 10.140

Find the characteristic function of the random variable $\tau (1)$ defined in problem 10.137.

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Problem 10.141

Find the probability that the Wiener process $w_{t}$ does not vanish on the interval $\left(t_{0}, t_{1}\right), 0 < t_{0} < t_{1}$ .

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Problem 10.142

Let $w_{t}$ be a Wiener process. Find the probability of the event

\left\{ \max _{0 \leqslant s \leqslant t} w_{s} \geqslant z, w_{t} < x\right\} .

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Problem 10.143

Let $w_{t}$ be a Wiener process. Find the distribution function and density of the random variable $\max_{0 \leqslant s \leqslant 1} w_{s}$ , given that $w_{1} = 0$ .

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Problem 10.144

Find the probability that the Wiener process $w_{t}$ reaches the sloped boundary given, in the coordinates $(t, w)$ , by the equation $w = a(t+1), t \geqslant 0, a > 0$ .

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Problem 10.145

(a)(b)

Let $P(a, b)$ denote the probability that the Wiener process $w_{t}$ reaches the sloped boundary given, in the coordinates $(t, w)$ , by the equation $w = a t+b, t \geqslant 0, a, b > 0$ . Prove that:

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(a)

$P(a, b) = P(b, a)$ ;

(b)

$P\left(a, b_{1}+b_{2}\right) = P\left(a, b_{1}\right) P\left(a, b_{2}\right)$ .

Problem 10.146

Let $P(a, b)$ be the quantity defined in the previous problem. Prove that

P(a, b) = e^{-\gamma a b},

where $\gamma$ is some nonnegative constant.

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Problem 10.147

Determine the value of the constant $\gamma$ in the previous problem.

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Problem 10.148

Prove that every process with independent increments is Markov.

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Problem 10.149

Let $\xi_{t}^{(1)}$ and $\xi_{t}^{(2)}, t \geqslant 0$ , be independent random processes, each of which is a process with independent increments. Prove that their sum

\xi _{t} = \xi _{t}^{(1)}+\xi _{t}^{(2)}, \quad t \geqslant 0,

is also a process with independent increments.

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Problem 10.150

Let $\xi_{t}$ be a random process with independent increments, $t \in R$ . Prove that if, for some $t_{1}$ and $t_{2}$ and some constant $a$ ,

\mathbb {P}\left(\xi _{t_{1}}-\xi _{t_{2}} = a\right) = 1,

then for any pair $u_{1}$ and $u_{2}$ such that $t_{1} \leqslant u_{1} \leqslant u_{2} \leqslant t_{2}$ , there exists a constant $b$ such that

\mathbb {P}\left(\xi _{u_{1}}-\xi _{u_{2}} = b\right) = 1.

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Problem 10.151

Prove that the distribution function of the increment of any homogeneous random process with independent increments is infinitely divisible.

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Problem 10.152

Let $\varphi (t, z)$ be the characteristic function of a homogeneous stochastically continuous process with independent increments $\xi_{t}$ . Prove that $\varphi (t, z)$ is continuous as a function of $t$ .

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Problem 10.153

Let $\xi_{t}$ be a process with independent increments, $\varphi (t, z)$ its characteristic function. Prove that if $\varphi (t, z)$ is continuous in $t$ at the point $t_{0}$ , then $\xi_{t}$ is stochastically continuous at the point $t_{0}$ .

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Problem 10.154

Let $\xi_{1}, \ldots , \xi_{n}$ be independent random variables, $t_{1} < t_{2} < \ldots < t_{n}$ points of the interval $[a, b)$ . Set $\xi_{t} = \sum_{t_{k} < t} \xi_{k}$ . Prove that $\xi_{t}$ is a process with independent increments.

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Problem 10.155

Let $\xi_{t}$ be a process with independent increments, $\varphi (t, z)$ its characteristic function. Prove that for each $z$ , $\left|\varphi (t, z)\right|$ is nonincreasing as a function of $t$ .

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Problem 10.156

Let $\xi_{t}$ be a homogeneous random process with independent increments, $\xi_{0} = 0$ , $\varphi (t, z)$ its characteristic function. Prove that for any $t$ and $s$ ,

\varphi (t+s, z) = \varphi (t, z) \varphi (s, z) .

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Problem 10.157

Let $\xi_{t}$ be a process with independent increments. Prove that if $\xi_{t_{0}}$ has an absolutely continuous distribution for some $t_{0}$ , then $\xi_{t}$ has an absolutely continuous distribution for every $t \geqslant t_{0}$ .

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Problem 10.158

Let $\xi_{t}$ be a process with independent increments. Prove that the function $\operatorname {Var}\left[\xi_{t}\right]$ is nondecreasing in $t$ .

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Problem 10.159

Let $\xi_{t}, a \leqslant t \leqslant b$ , be a homogeneous process with independent increments. Prove that $\xi_{t}$ is stochastically continuous everywhere on $[a, b]$ .

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Problem 10.160

Let $\xi_{t}, t \geqslant 0$ , be a homogeneous random process with independent increments, $\varphi_{\lambda }(u)$ the characteristic function of the random variable $\xi_{\lambda }-\xi_{0}$ . Prove that

\varphi _{\lambda }(u) = \left\{ \varphi _{1}(u)\right\} ^{\lambda } .

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Problem 10.161

Let $\xi_{t}$ be a process with independent increments, $\eta$ some random variable defined on the same probability space as $\xi_{t}$ . Is the process $\zeta_{t} = \xi_{t}+\eta$ a process with independent increments?

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Problem 10.162

Let $\xi_{t}, t \geqslant 0$ , be a homogeneous process with independent increments, not almost surely equal to a constant. Prove that $\xi_{t}$ is not stochastically bounded.

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Problem 10.163

Let $\xi_{t}, t \geqslant 0$ , be a process with independent increments. Prove that if for some $t_{0}$ , $\mathbb {P}\left(\xi_{t_{0}} = \text{ const }\right) = 1$ , then $\mathbb {P}\left(\xi_{t} = \text{ const }\right) = 1$ for all $t \leqslant t_{0}$ .

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Problem 10.164

Let $\xi_{t}, a \leqslant t \leqslant b$ , be a symmetric process with independent increments. Prove that if for some sequence $t_{1}, t_{2}, \ldots$ such that $t_{n} \rightarrow b$ as $n \rightarrow \infty$ and $t_{n} < b$ , $n = 1,2, \ldots$ , $\xi_{t_{n}} \rightarrow \xi_{b}$ , $n \rightarrow \infty$ , then

\xi _{t} \xrightarrow {\mathbb {P}} \xi _{b}, \quad t \rightarrow b .

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Problem 10.165

(Continuation) Can the symmetry condition be dropped?

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Problem 10.166

Let $\xi_{t}, a \leqslant t \leqslant b$ , be a process with independent increments. Is the process $\eta_{t} = \xi_{-t},-b \leqslant t \leqslant -a$ , a process with independent increments?

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Problem 10.167

Prove that if $\xi_{t}$ is a homogeneous process with independent increments, then there exists a positive constant $c$ ( $c$ may equal $+\infty$ ) such that $\operatorname {Var}\left[\xi_{t}\right] = c t$ .

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Problem 10.168

Let $\xi_{t}, a \leqslant t < b, 0 \leqslant a < b$ , be a process with independent increments. Can it be extended to the segments $[0, a)$ and $[b, \infty )$ so that the resulting process (on $[0, \infty )$ ) is also a process with independent increments?

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Problem 10.169

Let $\xi_{t}$ be a process with independent increments. Prove that if the function $\operatorname {Var}\left[\xi_{t}\right]$ is continuous in $t$ , then $\xi_{t}$ is stochastically continuous.

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Problem 10.170

Let $\xi_{t}, t \geqslant 0$ , be a homogeneous nondegenerate process with independent increments. Prove that for every $t > 0$ and every $A > 0$ , $\mathbb {P}\left(\left|\xi_{t}\right| > A\right) > 0$ .

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Problem 10.171

Let $\xi_{t}, t \geqslant 0$ , be a random process such that $\xi_{0}$ is uniformly distributed on the segment $[0,1]$ , and $\xi_{t_{0}}, t_{0} > 0$ , has an exponential distribution. Prove that $\xi_{t}$ cannot be a process with independent increments.

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Problem 10.172

Prove that a Gaussian random process with uncorrelated increments is a process with independent increments.

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Problem 10.173

Prove that any process with uncorrelated increments and mean zero has mean-square limits from the left and from the right at every point $t \in T$ .

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Problem 10.174

Let $\xi_{t}$ be a random process with uncorrelated increments, $t \in R$ . Prove that there exists a nondecreasing function $F(t)$ such that for any $t$ and $s$ , the random variable $\xi_{t}-\xi_{s}$ has variance equal to $F(t)-F(s)$ .

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Problem 10.175

Let $A, \eta$ and $\varphi$ be random variables, where $A, \eta$ are nonnegative and have an arbitrary joint distribution, and $\varphi$ is independent of them and has a uniform distribution on $[0,2 \pi )$ . Prove that the random process

\xi _{t} = A \cos (\eta t+\varphi ), t \in R,

is stationary.

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Problem 10.176

Let $\xi_{1}$ and $\xi_{2}$ be independent identically distributed random variables, taking the values $+1$ and $-1$ with probability $1/2$ each. Prove that the random process

\eta _{t} = \xi _{1} \cos \lambda t+\xi _{2} \sin \lambda t, \quad t \in R,

is not stationary, but is stationary in the wide sense.

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Problem 10.177

Let $\xi_{t}$ be a real Gaussian stationary process with mean zero and continuous correlation function $K(t)$ . Find the correlation function of the process $\eta_{s} = \xi_{t+s} \xi_{s}$ .

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Problem 10.178

Let $\pi_{t}, t \geqslant 0$ , be a Poisson process with parameter $\lambda$ . Prove that the process $\xi_{t} = \pi_{t+1}-\pi_{t}, t \geqslant 1$ , is stationary in the wide sense.

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Problem 10.179

Prove that if $\xi_{t}$ is a stationary process and there exists a limit of $\xi_{t}$ in probability as $t \rightarrow \infty$ , then for any $t_{1}, t_{2}$ , $\mathbb {P}\left(\xi_{t_{1}} = \xi_{t_{2}}\right) = 1$ .

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Problem 10.180

Let $T$ be a measure-preserving transformation and $\xi = \xi (\omega )$ a random variable with finite expectation. Prove that

\mathbb {E}\left[\xi (\omega )\right] = \mathbb {E}\left[\xi (T \omega )\right] .

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Problem 10.181

Let $\Omega = \left(\omega_{1}, \ldots , \omega_{n}\right)$ be a set consisting of finitely many points, $n \geqslant 2$ , $\mathscr {A}$ the set of all subsets, $T \omega_{i} = \omega_{i+1}, 1 \leqslant i \leqslant n-1$ , and $T \omega_{n} = \omega_{1}$ . Prove that if $\mathbb {P}\left(\omega_{i}\right) = 1/n$ , then $T$ is a measure-preserving transformation.

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Problem 10.182

(a)(b)

Let $(\Omega , \mathscr {B}, P)$ be the probability space where $\Omega = [0,1]$ , $\mathscr {B}$ the $\sigma$ -algebra of Borel sets, $P$ the Lebesgue measure. Let $\lambda \in [0,1)$ and

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(a)

$T(x) = (x+\lambda ) \bmod 1$ ;

(b)

$T(x) = 2 x \bmod 1$ .

Prove that $T$ is a measure-preserving transformation.

Problem 10.183

Let $\Omega = [0,1), \mathscr {A}$ the set of Borel subsets of $[0,1)$ , $P$ some measure with a continuous distribution function. Show that the transformations $T x = \lambda x, 0 < \lambda < 1$ , and $T x = x^{2}$ are not measure-preserving transformations.

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Problem 10.184

Prove that the class of sets invariant with respect to a measure-preserving transformation forms a $\sigma$ -algebra.

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Problem 10.185

Consider the same probability space as in problem 10.182. Prove that the transformation $T \omega = (\omega +\lambda ) \bmod 1$ is ergodic if and only if $\lambda$ is irrational.

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Problem 10.186

Prove that a random variable is invariant with respect to some measure-preserving transformation if and only if it is measurable with respect to the $\sigma$ -algebra of invariant events.

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Problem 10.187

Show that an event $A$ is invariant with respect to $T$ if and only if

\mathbb {P}\left(T^{-1} A \backslash A\right) = 0 \text{ or } \mathbb {P}\left(A \backslash T^{-1} A\right) = 0.

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Problem 10.188

Does the transformation considered in problem 10.185 have the mixing property?

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Problem 10.189

Prove that a transformation $T$ is mixing if and only if for any two random variables $\xi$ and $\eta$ having finite variances,

\mathbb {E}\left[\xi \left(T^{n} \omega \right) \eta (\omega )\right] \rightarrow \mathbb {E}\left[\xi \right] \mathbb {E}\left[\eta \right], \quad n \rightarrow \infty

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Problem 10.190

Is a sequence of pairwise independent identically distributed random variables stationary?

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Problem 10.191

Let $\xi_{1}, \xi_{2}, \ldots$ be a sequence of identically distributed random variables, where $\xi_{i}$ is independent of $\xi_{i-1}$ and $\xi_{i+1}$ . Is this sequence stationary?

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Problem 10.192

Let $\bar{\xi } = \left(\xi_{1}, \xi_{2}, \ldots \right)$ be a Gaussian stationary sequence with $\mathbb {E}\left[\xi_{n}\right] = 0$ and covariance function $R(n) = \mathbb {E}\left[\xi_{k+n} \xi_{k}\right]$ . Prove that the condition $R(n) \rightarrow 0$ is sufficient for the ergodicity of $\xi$ .

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Problem 10.193

Prove that every sequence consisting of independent identically distributed random variables is ergodic.

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Problem 10.194

Show that a stationary sequence $\xi = \left(\xi_{1}, \xi_{2}, \ldots \right)$ is ergodic if and only if for every $B \in \mathscr {B}\left(R^{k}\right), k = 1,2, \ldots$ ,

\frac{1}{n} \sum _{i = 1}^{n} I_{B}\left(\xi _{i}, \ldots , \xi _{i+k}\right) \xrightarrow {\text{a.s.}} \mathbb {P}\left(\left(\xi _{1}, \ldots , \xi _{k+1}\right) \in B\right) .

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Problem 10.195

State conditions under which a homogeneous Markov chain is a stationary sequence.

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Problem 10.196

Let $f\left(x_{0}, \ldots , x_{m}\right)$ be a measurable real function defined on $R^{m+1}$ , and $\left\{ \xi_{n}\right\}$ a stationary sequence of random variables. Prove that the sequence $\left\{ \eta_{n}\right\}$ , where $\eta_{n} = f\left(\xi_{n}, \ldots , \xi_{n+m}\right)$ , is also stationary.

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Problem 10.197

Prove that for a sequence of random variables $\left(\xi_{n}\right)$ to be stationary, it is necessary and sufficient that for every $m \geqslant 0$ and every bounded measurable function $f\left(x_{0}, \ldots , x_{m}\right)$ , $\mathbb {E}\left[f\left(\xi_{n}, \ldots , \xi_{n+m}\right)\right]$ not depend on $n$ .

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Problem 10.198

Prove that the ergodicity of the sequence $\left\{ \xi_{k}, k \geqslant 0\right\}$ implies the ergodicity of the sequence $\left\{ \eta_{k}, k \geqslant 0\right\}$ , where $\eta_{k} = f\left(\xi_{k}, \ldots , \xi_{k+m}\right)$ , and $f\left(x_{0}, \ldots , x_{m}\right)$ is an arbitrary measurable function.

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Problem 10.199

Let $\left\{ \xi_{k}\right\}$ be a stationary sequence, and $f_{n}\left(x_{0}, \ldots , x_{n}\right)$ a sequence of functions such that $f_{n}\left(\xi_{0}, \ldots , \xi_{n}\right)$ converges in probability to some random variable $\eta_{0}$ . Set $\quad \eta_{k} = \lim_{n \rightarrow \infty } f_{n}\left(\xi_{k}, \ldots , \xi_{k+n}\right)$ . Prove that $\left\{ \eta_{k}\right\}$ is a stationary sequence, and it is ergodic if $\left\{ \xi_{k}\right\}$ is ergodic.

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Problem 10.200

Prove that the sequence $\left\{ \xi_{k}\right\}$ is ergodic if and only if for every bounded measurable function $f\left(x_{0}, \ldots , x_{m}\right)$ ,

\lim _{n \rightarrow \infty } \frac{1}{n} \sum _{k = 0}^{n-1} f\left(\xi _{k}, \ldots , \xi _{k+m}\right) = \mathbb {E}\left[f\left(\xi _{0}, \ldots , \xi _{m}\right)\right] .

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Problem 10.201

Let $\left\{ \xi_{k}\right\}$ be a stationary sequence, $\mathbb {E}\left[\xi_{0} \xi_{n}\right]-\left(\mathbb {E}\left[\xi_{0}\right]\right)^{2} \rightarrow 0$ as $n \rightarrow \infty$ . Prove that

\lim _{n \rightarrow \infty } \frac{1}{n} \sum _{k = 0}^{n-1} \xi _{k} = \mathbb {E}\left[\xi _{0}\right] .

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Problem 10.202

Let $\left\{ \xi_{k}\right\}_{k \geq 0}$ be a Gaussian sequence. Prove that for this sequence to be stationary, it is necessary and sufficient that the equalities

\mathbb {E}\left[\xi _{n}\right] = \mathbb {E}\left[\xi _{0}\right], \quad n > 0, \quad \mathbb {E}\left[\xi _{0} \xi _{n}\right] = \mathbb {E}\left[\xi _{k} \xi _{k+n}\right], \quad k \geqslant 0, \quad n \geqslant 0,

hold.

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Problem 10.203

Let $\xi_{t}$ be a homogeneous process with independent increments. Prove that for $h > 0$ the process $\zeta_{t}^{(h)} = \xi_{t+h}-\xi_{t}$ is stationary.

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Problem 10.204

Let $\varphi (t)$ be a continuous periodic function with period $T$ , and $\xi$ a random variable uniformly distributed on the segment $[0, T]$ . Show that the process $\xi_{t} = \varphi (t+\xi )$ is stationary.

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Problem 10.205

Prove that the sum of independent stationary random processes is a stationary random process.

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Problem 10.206

Let $\xi_{1}, \ldots , \xi_{n}, \theta_{1}, \ldots , \theta_{n}$ be independent random variables, with $\theta_{1}, \ldots , \theta_{n}$ uniformly distributed on the segment $[0,2 \pi ]$ . Prove that the process

\xi _{t} = \sum _{k = 1}^{n} \xi _{k} \cos \left[k\left(\theta _{k}+t\right)\right]

is stationary.

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Problem 10.207

Let $\xi_{t}, t \geqslant 0$ , be a Gaussian process. Prove that it is stationary if and only if $\mathbb {E}\left[\xi_{t}\right] = \mathbb {E}\left[\xi_{0}\right]$ , $\mathbb {E}\left[\xi_{t} \xi_{t+s}\right] = \mathbb {E}\left[\xi_{0} \xi_{s}\right]$ for all $t \geqslant 0, s \geqslant 0$ .

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Problem 10.208

Let the correlation function of a random process $\xi_{t}$ stationary in the wide sense tend to zero at infinity. Prove that $\frac{1}{t_{2}-t_{1}} \int_{t_{1}}^{t_{2}} \xi_{t} d t$ converges to $\mathbb {E}\left[\xi_{t}\right]$ in the mean square as $t_{2}-t_{1} \rightarrow \infty$ .

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Problem 10.209

Prove that if $\xi_{t}$ is a mean-square continuous stationary process, $\mathbb {E}\left[\xi_{t}\right] \neq 0$ , then there is no random variable $\eta$ such that $\eta +\int_{0}^{t} \xi_{s} d s$ is a stationary process.

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Problem 10.210

Let $\xi_{t}$ be a stationary process, $\eta$ a random variable. Is the process $\zeta_{t} = \xi_{t}+\eta$ stationary?

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Problem 10.211

Find the spectral representation of the random process $\xi_{t}$ defined in problem 10.175.

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Problem 10.212

Let $\xi_{t}$ be a real stationary process with mean $m$ and spectral density $f(\lambda )$ . Set $\eta_{t} = \xi_{t} \cos (\Lambda t+\varphi )$ , where $\Lambda = \text{const}$ , and $\varphi$ is a random variable, independent of $\xi_{t}$ , uniformly distributed on $[0,2 \pi )$ . Find the spectral representation for $\eta_{t}$ .

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Problem 10.213

Let $\xi_{0}, \xi_{1}, \ldots$ be a sequence of independent random variables with mean zero, $S_{n} = \xi_{0}+\ldots +\xi_{n}$ . Prove that the sequence $\left\{ S_{n}\right\}$ forms a martingale.

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Problem 10.214

Let $\xi_{0}, \xi_{1}, \ldots$ be a sequence of independent random variables, $\mathbb {E}\left[\xi_{n}\right] = 1, n = 0,1, \ldots , \quad X_{n} = \prod_{i = 0}^{n} \xi_{i}$ . Prove that the sequence $\left\{ X_{n}\right\}$ forms a martingale.

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Problem 10.215

Let $\xi$ be a random variable with finite expectation, $\left\{ \mathscr {F}_{n}\right\}_{n \geq 0}$ a nondecreasing sequence of $\sigma$ -algebras. Set $\xi_{n} = \mathbb {E}\left[ \xi \mid \mathscr {F}_{n} \right]$ . Prove that the sequence $\left\{ \xi_{n}, \mathscr {F}_{n}\right\}$ forms a martingale.

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Problem 10.216

Let $\left\{ \xi_{k}\right\}$ be a sequence of independent random variables, and $\left\{ \eta_{k}\right\}$ a sequence of random variables such that for each $k$ , $\left\{ \eta_{1}, \ldots , \eta_{k}\right\}$ and $\left\{ \xi_{k}, \xi_{k+1}, \ldots \right\}$ are independent collections of random variables. Prove that if $\mathbb {E}\left[\xi_{k}\right] = 0, \mathbb {E}\left[\left|\eta_{k} \xi_{k}\right|\right] < \infty$ , then the sequence

\zeta _{n} = \sum _{k = 1}^{n} \eta _{k} \xi _{k}, \quad n = 1,2, \ldots

is a martingale.

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Problem 10.217

Let $\left\{ \xi_{n}\right\}$ be a martingale, $\mathbb {E}\left[\xi_{n}^{2}\right] < \infty$ . Prove that $\left\{ \xi_{n}^{2}\right\}$ is a submartingale.

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Problem 10.218

Let $\left\{ \xi_{n}\right\}$ be a sequence of nonnegative random variables having finite expectations, $S_{n} = \xi_{0}+\ldots +\xi_{n}$ . Prove that the sequence $\left\{ S_{n}\right\}$ forms a submartingale.

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Problem 10.219

Let $\left\{ X_{n}, \mathscr {F}_{n}\right\}$ be a martingale, and $g(x)$ a convex function such that $\mathbb {E}\left[\left|g\left(X_{n}\right)\right|\right] < \infty , n = 0,1, \ldots$ Prove that the sequence $\left\{ g\left(X_{n}\right), \mathscr {F}_{n}\right\}$ forms a submartingale.

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Problem 10.220

Let $\left\{ \xi_{n}\right\}$ and $\left\{ \eta_{n}\right\}$ be two sequences of random variables such that for each $n$ there exist a joint density of the random variables $\xi_{1}, \ldots , \xi_{n}$ , $f_{n}\left(x_{1}, \ldots , x_{n}\right)$ , and a joint density of the random variables $\eta_{1}, \ldots , \eta_{n}$ , $g_{n}\left(x_{1}, \ldots , x_{n}\right)$ . Prove that the sequence

\zeta _{n} = \frac{g_{n}\left(\xi _{1}, \ldots , \xi _{n}\right)}{f_{n}\left(\xi _{1}, \ldots , \xi _{n}\right)}, \quad n = 1,2, \ldots

forms a martingale.

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Problem 10.221

Let $\left\{ X_{n}, \mathscr {F}_{n}\right\}$ be a submartingale, and $g(x)$ a convex nondecreasing function such that $\mathbb {E}\left[\left|g\left(X_{n}\right)\right|\right] < \infty , n = 0,1, \ldots$ Prove that the sequence $\left\{ g\left(X_{n}\right), \mathscr {F}_{n}\right\}$ also forms a submartingale.

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Problem 10.222

Let $\left\{ \mathscr {F}_{n}\right\}$ be a nondecreasing sequence of $\sigma$ -algebras, $\left\{ X_{n}\right\}$ a sequence of random variables such that $X_{n}$ is measurable with respect to $\mathscr {F}_{n}$ . Let $B$ be an arbitrary Borel set on the line. Prove that the first hitting time of the set $B$ : $\tau_{B} = \inf \left\{ n \geqslant 0: X_{n} \in B\right\}$ is a Markov time.

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Problem 10.223

Let $\tau$ and $\sigma$ be Markov times. Prove that $\tau +\sigma$ , $\min \left\{ \tau , \sigma \right\} , \max \left\{ \tau , \sigma \right\}$ are also Markov times with respect to the same sequence of $\sigma$ -algebras as $\tau , \sigma$ .

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Problem 10.224

Let $\tau$ and $\sigma$ be Markov times. Is the random variable $\tau -\sigma$ a Markov time?

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Problem 10.225

Let $\left\{ X_{n}, \mathscr {F}_{n}\right\}$ be a martingale (submartingale), $\tau$ a Markov time with respect to the sequence of $\sigma$ -algebras $\left\{ \mathscr {F}_{n}\right\}$ . Set $\tau_{n} = \min \left\{ n, \tau \right\}$ . Prove that the sequence $\left\{ X_{\tau_{n}}, \mathscr {F}_{n}\right\}$ is also a martingale (submartingale).

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Problem 10.226

Let $\left\{ \xi_{n}\right\}_{n > 0}$ be a sequence of independent identically distributed random variables, $\mathbb {P}\left(\xi_{i} = 0\right) = \mathbb {P}\left(\xi_{i} = 2\right) = 1/2$ , $X_{n} = \prod_{i = 1}^{n} \xi_{i}$ . Show that there is no integrable random variable $\xi$ and nondecreasing family of $\sigma$ -algebras $\left\{ \mathscr {F}_{n}\right\}$ such that $X_{n} = \mathbb {E}\left[ \xi \mid \mathscr {F}_{n} \right]$ .

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Problem 10.227

Let $\xi_{t}$ be a homogeneous Poisson process with parameter $\lambda$ . Prove that $\zeta_{t} = \exp \left\{ \xi_{t}-a t\right\}$ is a submartingale for $a \leqslant \lambda (e-1)$ and a supermartingale for $a \geqslant \lambda (e-1)$ .

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Problem 10.228

Let $\left\{ \mathscr {F}_{n}\right\}$ be a nondecreasing sequence of $\sigma$ -algebras, $\mathscr {F}$ the $\sigma$ -algebra generated by $\bigcup \mathscr {F}_{n}$ . Let $\xi$ be a nonnegative random variable, measurable with respect to $\mathscr {F}$ , having finite expectation. Prove that $\lim_{n \rightarrow \infty } \mathbb {E}\left[\xi \mid \mathscr {F}_{n}\right] = \xi$ .

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Problem 10.229

Let $\left\{ \xi_{n}\right\}$ be a uniformly integrable martingale, $\xi_{n} \geqslant 0, \xi = \lim \xi_{n}$ . Prove that $\xi_{n} = \mathbb {E}\left[ \xi \mid \mathscr {F}_{n} \right]$ , where $\mathscr {F}_{n}$ is the $\sigma$ -algebra generated by the variables $\xi_{1}, \ldots , \xi_{n}$ .

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Problem 10.230

Let $\left\{ X_{n}\right\} , n \geqslant 0$ , be a martingale. Prove that $\mathbb {E}\left[X_{n}\right] = \mathbb {E}\left[X_{0}\right]$ for every $n = 1,2, \ldots$

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Problem 10.231

Prove that a Wiener process started from zero is a martingale.

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Problem 10.232

Prove that if $\xi_{t}$ ( $t \geqslant 0$ ) is a process with independent increments, $\xi_{0} = 0$ , and $\mathbb {E}\left[\xi_{t}\right] = 0$ for every $t \geqslant 0$ , then $\xi_{t}$ forms a martingale.

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Problem 10.233

Let $\xi_{t}, t \in T$ , be a martingale with respect to the family $\mathscr {F}_{t}$ , $\mathbb {E}\left[\left|\xi_{t}\right|^{2}\right] < \infty$ . Prove that $\xi_{t}$ has uncorrelated increments.

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Problem 10.234

Let $\xi_{t}, t \geqslant 0$ , be a process with independent increments, $\xi_{0} = 0, \mathbb {E}\left[\xi_{t}\right] = 0, \mathbb {E}\left[\left(\xi_{t}-\xi_{s}\right)^{2}\right] = F(t)-F(s), s \leqslant t$ . Prove that $\left(\xi_{t}^{2}-F(t), \mathscr {F}_{t}\right)$ is a martingale, where $\mathscr {F}_{t} = \sigma \left(\xi_{u}, u \leqslant t\right)$ .

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Problem 10.235

Let $\left\{ \xi_{n}\right\}$ be a martingale and $\mathbb {E}\left[\xi_{n}^{2}\right] < \infty , n = 1,2, \ldots$ Prove that for every $a > 0$ ,

\mathbb {P}\left(\sup _{1 \leqslant k \leqslant n}\left|\xi _{k}\right| \geqslant a\right) \leqslant \frac{\mathbb {E}\left[\xi _{n}^{2}\right]}{a^{2}} .

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Problem 10.236

Let $\left\{ \xi_{n}\right\}$ be a submartingale. Prove that $\xi_{n} = \eta_{n}+\zeta_{n}$ , where $\left\{ \eta_{n}\right\}$ is a martingale, and $\zeta_{n}$ is measurable with respect to $\sigma \left(\xi_{1}, \ldots , \xi_{n-1}\right)$ and $\mathbb {P}\left(0 \leqslant \zeta_{1} \leqslant \zeta_{2} \leqslant \ldots \right) = 1$ .

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Problem 10.237

Let $\left\{ \xi_{n}\right\}$ be a submartingale. Prove that for every $a > 0$ ,

\mathbb {P}\left(\sup _{1 \leqslant k \leqslant n} \xi _{k} \geqslant a\right) \leqslant \frac{\mathbb {E}\left[\max \left\{ 0, \xi _{n}\right\} \right]}{a}

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Problem 10.238

Let $\left\{ \xi_{n}\right\}$ be a supermartingale. Prove that for every $a \geqslant 0$ ,

\mathbb {P}\left(\sup _{1 \leqslant k \leqslant n} \xi _{k} \geqslant a\right) \leqslant \frac{1}{a}\left(\mathbb {E}\left[\max \left\{ 0, \xi _{n}\right\} \right]-\mathbb {E}\left[\xi _{n}\right]+\mathbb {E}\left[\xi _{1}\right]\right) .

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Problem 10.239

Let $\left\{ \xi_{n}\right\}$ be a submartingale. Prove that for every $a \geqslant 0$ ,

\mathbb {P}\left(\sup _{1 \leqslant k \leqslant n}\left|\xi _{k}\right| \geqslant a\right) \geqslant \frac{1}{a}\left(\mathbb {E}\left[\left|\xi _{n}\right|\right]-\mathbb {E}\left[\xi _{1}\right]+\mathbb {E}\left[\xi _{n}\right]\right) .

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Problem 10.240

Let $\left\{ \xi_{n}\right\}$ be a nonnegative submartingale, $\quad \mathbb {E}\left[\xi_{n}^{2}\right] \leqslant C$ , $n = 1,2, \ldots$ Prove that the limit $\lim_{n \rightarrow \infty } \xi_{n}$ exists almost surely.

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Problem 10.241

Let $\left\{ \xi_{n}\right\}$ be an arbitrary submartingale, $\mathbb {E}\left[\xi_{n}^{2}\right] \leqslant C$ , $n = 1,2, \ldots$ Prove that the limit $\lim_{n \rightarrow \infty } \xi_{n}$ exists almost surely.

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Problem 10.242

Let $\left\{ \xi_{n}\right\}$ be a nonnegative martingale. Prove that the limit $\lim_{n \rightarrow \infty } \xi_{n}$ exists almost surely.

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Problem 10.243

Find the transition function of a Poisson process, regarding it as a Markov process.

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Problem 10.244

Let $w_{t}, t \geqslant 0$ , be a Wiener process. Find the transition function of the process $\xi_{t} = w_{-t}, t \leqslant 0$ , regarding it as a Markov process.

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Problem 10.245

Let $w_{t}, t \geqslant 0$ , be a Wiener process. Prove that $\left|w_{t}\right|$ is a Markov process. Find its transition function.

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Problem 10.246

Let $\pi_{t}$ be a Poisson process. Find $\mathbb {E}\left[\left[\pi_{t}-\pi_{s}\right]^{n}\right]$ , $n = 1,2, \ldots$

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Problem 10.247

Let $\pi_{t}, t \geqslant 0$ , be a Poisson process with parameter $\lambda$ . Prove that the distribution of $\frac{\pi_{t}-\lambda t}{\sqrt{\lambda t}}$ converges weakly, as $t \rightarrow \infty$ , to the normal distribution with parameters $0$ and $1$ .

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Problem 10.248

Show that if a stationary process is Gaussian and Markov, then its covariance function has the form $c e^{-\alpha \left|t\right|}, \alpha \geqslant 0, c > 0$ , for some constant.

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Problem 10.249

Let $w_{s}$ be a Wiener process, $\xi_{t} = e^{-t} w_{e^{2t}}$ . Show that $\xi_{t}$ is a stationary Markov process. Find its covariance function and spectral density.

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Problem 10.250

Let $\tau (a)$ be the first time the Wiener process $w_{t}$ reaches the level $a$ . Set

w_{t}^{a} = \begin{cases} w_{t}, & t \leqslant \tau (a) \\ a, & t > \tau (a) \end{cases}

Prove that $w_{t}^{a}$ is a Markov process. Find its transition function.

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Problem 10.251

Let $w_{t}$ be a Wiener process. Find the probability of the event $\left\{ \sup_{0 \leqslant s \leqslant t}\left|w_{s}\right| \geqslant z, a \leqslant w_{t} \leqslant b\right\}$ , where $z > 0$ and $-z < a < b < z$ .

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Problem 10.252

Let $w_{t}$ be a Wiener process. Find the distribution function of the random variable $\sup_{0 \leqslant t \leqslant 1}\left|w_{t}\right|$ , given that $w_{1} = 0$ .

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Problem 10.253

Let $w_{s}^{(1)}$ and $w_{s}^{(2)}$ be independent Wiener processes. For every real $t$ , set

\bar{w}_{t} = \begin{cases} w_{t}^{(1)}, & t \geqslant 0, \\ w_{-t}^{(2)}, & t < 0. \end{cases}

Further, let $\xi_{t} = \frac{1}{h}\left(\bar{w}_{t}-\bar{w}_{t-h}\right), \quad h = \text{const}$ . Show that $\xi_{t}$ is a stationary process. Find its covariance function and spectral density.

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Problem 10.254

Let $\ldots , \xi_{-2}, \xi_{-1}, \xi_{0}, \xi_{1}, \xi_{2}, \ldots$ be independent identically distributed random variables such that $\mathbb {E}\left[\xi_{k}\right] = 0$ , $\mathbb {E}\left[\xi_{k}^{2}\right] = \sigma^{2}$ . Set $\eta_{n} = \xi_{n}+\xi_{n-1}+\ldots +\xi_{n-m}$ ( $n = 0, \pm 1, \ldots$ , $m = \text{const}$ ). Show that the sequence $\left\{ \eta_{n}\right\}$ forms a stationary process. Find the covariance function of this process.

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Problem 10.255

Let $\ldots , \xi_{-2}, \xi_{-1}, \xi_{0}, \xi_{1}, \xi_{2}, \ldots$ be independent identically distributed random variables such that $\quad \mathbb {E}\left[\xi_{h}\right] = 0$ , $\mathbb {E}\left[\xi_{h}^{2}\right] = \sigma^{2}$ . Set $\eta_{n} = c_{0} \xi_{n}+\ldots +c_{m} \xi_{n-m}, n = 0, \pm 1, \ldots$ , where $c_{0}, \ldots , c_{m}$ are arbitrary real numbers, $m$ fixed. Show that the sequence $\left\{ \eta_{n}\right\}$ forms a stationary process, and find its covariance function.

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Problem 10.256

Show that the function $R(t) = \sigma^{2} e^{-\alpha \left|t\right|} \cos \beta t$ , where $\alpha , \beta$ , and $\sigma$ are some positive constants, can be the covariance function of a continuous process, stationary in the wide sense. Determine the spectral density corresponding to such a covariance function.

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Problem 10.257

Let $X_{0}, X_{1}, \ldots$ be a branching process, $X_{0} = 1, \mathbb {E}\left[X_{1}\right] = m$ , $Z_{n} = X_{n} / m^{n}$ . Prove that $\left\{ Z_{n}\right\}$ forms a martingale.

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