Stochastic Processes. Distributions of Stochastic Processes

Using the Kolmogorov theorem (Theorem 4), obtain a special case of the Lomnicki-Ulam theorem (Theorem 2) for a family of Borel spaces.

Theorem 4 (Kolmogorov). Let $(S_{t}, \mathscr {B}_{t})_{t \in T}$ be a family of Borel spaces (i.e., measurable spaces isomorphic to a Borel subset of the interval $[0,1]$ with its Borel $\sigma$-algebra). Suppose that on the spaces $(S_{t_{1}, \ldots , t_{n}}, \mathscr {B}_{t_{1}, \ldots , t_{n}})$, where $n \in \mathbb {N}$ and the points $t_{1}, \ldots , t_{n} \in T$ are pairwise distinct, measures $P_{t_{1}, \ldots , t_{n}}$ are given that satisfy, for every permutation $(i_{1}, \ldots , i_{n})$ of the tuple $(1, \ldots , n)$, the symmetry condition $P_{t_{1}, \ldots , t_{n}}\left(B_{t_{1}} \times \cdots \times B_{t_{n}}\right) = P_{t_{i_{1}}, \ldots , t_{i_{n}}}\left(B_{t_{i_{1}}} \times \cdots \times B_{t_{i_{n}}}\right)$ and the consistency condition $P_{t_{1}, \ldots , t_{n}}\left(B_{t_{1}} \times \cdots \times B_{t_{n - 1}} \times S_{t_{n}}\right) = P_{t_{1}, \ldots , t_{n - 1}}\left(B_{t_{1}} \times \cdots \times B_{t_{n - 1}}\right)$. Then there exist a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and a random function $X = \left\{ X(t), t \in T\right\}$ defined on it such that the finite-dimensional distributions of $X$ are the measures $P_{t_{1}, \ldots , t_{n}}$.

Theorem 2 (Lomnicki-Ulam). Let $(S_{t}, \mathscr {B}_{t})_{t \in T}$ be an arbitrary family of measurable spaces, and suppose that for every $t \in T$ a measure $Q_{t}$ is given on $(S_{t}, \mathscr {B}_{t})$. Then there exist a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and a family of independent random elements $X_{t}: \Omega \rightarrow S_{t}, X_{t} \in \mathscr {F} \mid \mathscr {B}_{t}$, such that $\mathbb {P}_{X_{t}} = Q_{t}$ on $\mathscr {B}_{t}$ for each $t \in T$.

Verify that the variables $a_{1}(\omega ), a_{2}(\omega ), \ldots$, introduced in (50), form a sequence of independent random variables on the indicated probability space $(\Omega , \mathscr {F}, \mathbb {P})$, and moreover one such that

Let $\Omega = [0,1], \mathscr {F} = \mathscr {B}([0,1])$ and $\mathbb {P} =$ mes (Lebesgue measure). Consider the binary expansion of the number $\omega \in \Omega$ :

where the coefficients $a_{k}(\omega )$ are equal to 0 or 1. Representation (50) is not unique. It is assumed that in the non-unique cases only the representations with infinitely many zeros among the coefficients $a_{k}(\omega )$ are considered.

Prove that if on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ there is a sequence of independent random variables satisfying condition (51), then the random variable $\xi (\omega ) = \sum_{k = 1}^{\infty } a_{k}(\omega ) 2^{-k}$ has the uniform distribution on the interval $[0,1]$.

Prove that if $F$ is some distribution function and $U$ is a random variable uniformly distributed on the interval $[0,1]$ (defined on some probability space), then $X(\omega ) = F^{\text{inv }}(U(\omega ))$ is a random variable with distribution function $F$ (the generalized inverse function $F^{\operatorname {inv}}(x) = \inf \left\{ y: F(y) \geqslant x\right\} , x \in [0,1]$; if $F(y) < 1$ for all $y \in \mathbb {R}$, then $F^{\text{inv }}(1): = +\infty$; if $F(y) > 0$ for all $y \in \mathbb {R}$, then $F^{\operatorname {inv}}(0): = -\infty$).

(compare with Theorem 6). Give an example of a family of $\sigma$-algebras $\mathscr {A}_{\lambda }, \lambda \in \Lambda$, consisting of some subsets of $\Omega$, such that $\bigcup_{\lambda \in \Lambda } \mathscr {A}_{\lambda }$ is not a $\sigma$-algebra.

Theorem 6. Let $T$ be an infinite set and $N(T)$ the collection of countable subsets of $T$. Then for the $\sigma$-algebra $\mathscr {B}_{T}$ the representations $\mathscr {B}_{T} = \sigma \left\{ \mathscr {C}_{T}\right\}$ and $\mathscr {B}_{T} = \bigcup_{U \in N(T)} \mathscr {B}_{T, U}$ hold, where $\mathscr {C}_{T}$ is the algebra of cylinder sets (the union of $\mathscr {B}_{T, J}$ over all finite $J \subset T$), and $\mathscr {B}_{T, U} = \pi_{T, U}^{-1} \mathscr {B}_{U}$ is the $\sigma$-algebra of sets of the space $S_{T}$ that depend only on the coordinates from $U$ (the preimage of $\mathscr {B}_{U}$ under the coordinate projection $\pi_{T, U}: S_{T} \rightarrow S_{U}$). In other words, whether a function $y \in S_{T}$ belongs to any set from $\mathscr {B}_{T}$ is determined only by its values on some countable subset of $T$.

Obtain a generalization of Lemma 7 to random elements with values in a Borel space.

Lemma 7. Let the probability space $(\Omega , \mathscr {F}, \mathbb {P})$ be complete, and let $\sigma$-algebra $\mathscr {A} \subset \mathscr {F}$. A real random variable $Y(\omega )$ is $\overline{\mathscr {A}}$-measurable (here $\overline{\mathscr {A}} = \sigma \left\{ \mathscr {A}, \mathscr {N}\right\}$, where $\mathscr {N}$ is the class of $\mathbb {P}$-null sets) if and only if there exists an $\mathscr {A}$-measurable real random variable $Z(\omega )$ such that $Y = Z$ ($\mathbb {P}$-a.e.), i.e., $\mathbb {P}\left(\omega \in \Omega : Y(\omega ) \neq Z(\omega )\right) = 0$.

Let $(\mathrm{S}_{t}, \mathscr {B}_{t})_{t \in T}$ be a family of measurable spaces, and let $\mathrm{S}_{T} = \prod_{t \in T} \mathrm{S}_{t}$ be their Cartesian product, consisting of the functions $y = y(t)$ defined on $T$ and such that $y(t) \in \mathrm{S}_{t}$ for $t \in T$.

Definition 6. The $\sigma$-algebra $\mathscr {B}_{T}$ in the space $\mathrm{S}_{T}$ generated by the collection of elementary cylinders, i.e., sets of the form $C_{T}(t, B_{t}) = \left\{ y \in \mathrm{S}_{T}: y(t) \in B_{t}\right\}$, where $B_{t} \in \mathscr {B}_{t}$, $t \in T$, is called the cylindrical $\sigma$-algebra.

Every random function $X = \left\{ X(t), t \in T\right\}$, regarded as a random element $\omega \mapsto X(\cdot , \omega )$ with values in $\mathrm{S}_{T}$, induces a probability distribution $\mathbb {P}_{X}$ on $(\mathrm{S}_{T}, \mathscr {B}_{T})$.

Let the set $G$ consist of the functions $g: \mathbb {R}^{d} \rightarrow \mathbb {R}$ that are nondecreasing in each argument. Show that $G \notin \mathscr {B}_{T}\left(\mathrm{S}_{t} = \mathbb {R}, t \in T = \mathbb {R}^{d}\right)$. Nevertheless, find $\mathbb {P}_{X}\left(G\right)$, where $X = \left\{ X_{t} = \xi (1+f(t) \eta ), t \in \mathbb {R}^{d}\right\} , f: \mathbb {R}^{d} \rightarrow \mathbb {R}$ is a function nonincreasing in each variable, and $\xi$ and $\eta$ are independent variables uniformly distributed on the interval $[-1,2]$.

(compare with Exercise 2). Prove that on $(\Omega , \mathscr {F}, \mathbb {P}) = ([0,1], \mathscr {B}([0,1]), \mathbb {P})$, where $\mathbb {P}$ is Lebesgue measure, it is impossible to construct a (continuum) family $\left\{ X_{t}, t \in \mathbb {R}\right\}$ of independent Bernoulli variables, i.e., random variables $X_{t}, t \in \mathbb {R}$, such that

Prove that on no probability space is it possible to define a family of independent identically distributed real nondegenerate random variables $X(t, \omega ), t \in \mathbb {R}, \omega \in \Omega$, in such a way that with probability 1 the sample paths $X(\cdot , \omega )$ are continuous on $\mathbb {R}$.

(compare with Theorem 4). Let $C(\mathbb {R})$ be the space of real functions continuous on $\mathbb {R}$. On the cylinder sets of the form $D_{J} = \left\{ x \in C(\mathbb {R}): x(t) \in B_{t}, t \in J\right\}$, where $B_{t} \in \mathscr {B}(\mathbb {R}), t \in J$ and $J \in F(\mathbb {R})$ ($F(\mathbb {R})$ is the collection of finite subsets of $\mathbb {R}$), define the functions

here $\mathbb {Q}_{0}$ is some measure on $\mathscr {B}(\mathbb {R})$. Is it possible to introduce a measure $\mathbb {P}$ on the $\sigma$-algebra of cylinder sets of the space $C(\mathbb {R})$ such that $\mathbb {P}\left(D_{J}\right) = \mathbb {Q}_{J}\left(D_{J}\right)$ for all $J \in F(\mathbb {R})$ and all cylinders $D_{J}$ introduced above?

Theorem 4 (Kolmogorov). Let $(S_{t}, \mathscr {B}_{t})_{t \in T}$ be a family of Borel spaces (i.e., measurable spaces isomorphic to a Borel subset of the interval $[0,1]$ with its Borel $\sigma$-algebra). Suppose that on the spaces $(S_{t_{1}, \ldots , t_{n}}, \mathscr {B}_{t_{1}, \ldots , t_{n}})$, where $n \in \mathbb {N}$ and the points $t_{1}, \ldots , t_{n} \in T$ are pairwise distinct, measures $P_{t_{1}, \ldots , t_{n}}$ are given that satisfy, for every permutation $(i_{1}, \ldots , i_{n})$ of the tuple $(1, \ldots , n)$, the symmetry condition $P_{t_{1}, \ldots , t_{n}}\left(B_{t_{1}} \times \cdots \times B_{t_{n}}\right) = P_{t_{i_{1}}, \ldots , t_{i_{n}}}\left(B_{t_{i_{1}}} \times \cdots \times B_{t_{i_{n}}}\right)$ and the consistency condition $P_{t_{1}, \ldots , t_{n}}\left(B_{t_{1}} \times \cdots \times B_{t_{n - 1}} \times S_{t_{n}}\right) = P_{t_{1}, \ldots , t_{n - 1}}\left(B_{t_{1}} \times \cdots \times B_{t_{n - 1}}\right)$. Then there exist a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and a random function $X = \left\{ X(t), t \in T\right\}$ defined on it such that the finite-dimensional distributions of $X$ are the measures $P_{t_{1}, \ldots , t_{n}}$.

What do the sample paths of process (11) look like?

Example 3. On some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ consider two sequences of nonnegative random variables $\left\{ \xi_{j}, j \in \mathbb {N}\right\}$ and $\left\{ \eta_{j}, j \in \mathbb {N}\right\}$ such that $\left\{ \xi_{j}, \eta_{j}, j \in \mathbb {N}\right\}$ are jointly independent and $\operatorname {Law}\left(\xi_{j}\right) = \operatorname {Law}\left(\xi_{1}\right)$, $\operatorname {Law}\left(\eta_{j}\right) = \operatorname {Law}\left(\eta_{1}\right)$, $j \in \mathbb {N}$. For positive constants $y_{0}$ and $c$ define the process

where $X_{t}(\omega )$ is given by formula (10).

Example 2. For a sequence $\xi_{1}, \xi_{2}, \ldots$ of independent identically distributed nondegenerate nonnegative random variables, the renewal process is defined by the formula

Throughout, the sum over the empty set is taken to be zero, so $X_{t}(\omega ) = 0$ if $\xi_{1}(\omega ) > t$.

Let $m = 1$ in definition (13), i.e., $\left\{ \zeta_{j}\right\}$ are independent identically distributed scalar random variables, and let $d = 1$.

Example 5. Let $\left\{ \zeta_{j}, j \in \mathbb {Z}^{d}\right\}$ be a random field formed by independent identically distributed vectors with values in $\mathbb {R}^{m}$. Let $\mu$ be a $\sigma$-finite measure on $\mathscr {B}\left(\mathbb {R}^{d}\right)$, $d \geqslant 1$. Define the partial-sum process

where $C_{j} = (j-1, j] = \left(j_{1}-1, j_{1}\right] \times \cdots \times \left(j_{d}-1, j_{d}\right]$ is the unit cube with upper vertex at the point $j \in \mathbb {Z}^{d}$, $n B = \left\{ x = n y, y \in B\right\}$.

Set

Sketch the graph of a sample path of this process in the following cases:

Let $m = 1$ in definition (12).

Example 4. Let $\xi_{1}, \xi_{2}, \ldots$ be independent identically distributed random vectors defined on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in $\mathbb {R}^{m}$ ($m \geqslant 1$). Introduce the empirical measures

Set

Sketch the graph of a sample path of the process $F_{n}^{*}(x, \omega )$, which is customarily called the empirical distribution function. What is the difference between the case when the $\xi_{j}$ are uniformly distributed on $[0,1]$ and the case when the $\xi_{j}$ are Bernoulli variables with $\mathbb {P}\left(\xi_{j} = 1\right) = p, \mathbb {P}\left(\xi_{j} = 0\right) = 1-p, 0 < p < 1 ?$

Example 4. Let $\xi_{1}, \xi_{2}, \ldots$ be independent identically distributed random vectors defined on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in $\mathbb {R}^{m}$ ($m \geqslant 1$). Introduce the empirical measures

Definition 15. A class of sets $\mathscr {M} \subset \mathscr {B}\left(\mathbb {R}^{m}\right)$ is called finitely approximable with respect to a measure $\mathbb {Q}$ defined on $\mathscr {B}\left(\mathbb {R}^{m}\right)$ if for every $\varepsilon > 0$ one can find sets $S_{1}^{(\varepsilon )}, \ldots , S_{N}^{(\varepsilon )} \in \mathscr {B}\left(\mathbb {R}^{m}\right)$, where $N = N(\varepsilon )$, such that for each $B \in \mathscr {M}$ there exist $S_{i}^{(\varepsilon )}$ and $S_{j}^{(\varepsilon )}$ with the properties $S_{i}^{(\varepsilon )} \subset B \subset S_{j}^{(\varepsilon )}$ and $\mathbb {Q}\left(S_{j}^{(\varepsilon )} \backslash S_{i}^{(\varepsilon )}\right) < \varepsilon$.

Prove that the class $\mathscr {M} = \left\{ (-\infty , x], x \in \mathbb {R}^{m}\right\}$, i.e., the collection of sets of the form $\left(-\infty , x_{1}\right] \times \cdots \times \left(-\infty , x_{m}\right], \left(x_{1}, \ldots , x_{m}\right) \in \mathbb {R}^{m}$, is finitely approximable with respect to any (probability) measure $\mathbb {Q}$. Thus the multidimensional version of the Glivenko-Cantelli theorem holds, on the a.s. uniform convergence as $n \rightarrow \infty$ of the empirical distribution functions $F_{n}^{*}(x, \omega ): = \mathbb {P}_{n}\left((-\infty , x], \omega \right)$ to the distribution function $F_{\xi_{1}}(x)$ :

Prove that if the measure $\mathbb {Q} = \mathbb {P}_{\xi_{1}}$ is absolutely continuous with respect to Lebesgue measure in $\mathbb {R}^{m}$, then statement (52) also holds for the class $\mathscr {M}$ that is the class of convex sets. (This statement, which follows from Exercise 13 when $m = 1$, becomes nontrivial and no longer follows from it if $m \geqslant 2$.)

Theorem 8 (see [3; p. 421]). Let $\mathscr {M}$ be a class of sets that is finitely approximable with respect to a measure $\mathbb {Q}$. Let $\mathbb {P}_{n}\left(B, \omega \right)$ be the process defined by formula (12), and let the corresponding vectors have the distribution law $\mathbb {P}_{\xi_{1}} = \mathbb {Q}$. Then

for all $\omega \in \widetilde{\Omega }$, where $\Omega \backslash \widetilde{\Omega } \subset \Omega_{0}$ and $\mathbb {P}\left(\Omega_{0}\right) = 0$. In other words, if the probability space $(\Omega , \mathscr {F}, \mathbb {P})$ is complete, then the convergence in (52) holds almost surely.

Example 4. Let $\xi_{1}, \xi_{2}, \ldots$ be independent identically distributed random vectors defined on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and taking values in $\mathbb {R}^{m}$ ($m \geqslant 1$). Introduce the empirical measures

Definition 15. A class of sets $\mathscr {M} \subset \mathscr {B}\left(\mathbb {R}^{m}\right)$ is called finitely approximable with respect to a measure $\mathbb {Q}$ defined on $\mathscr {B}\left(\mathbb {R}^{m}\right)$ if for every $\varepsilon > 0$ one can find sets $S_{1}^{(\varepsilon )}, \ldots , S_{N}^{(\varepsilon )} \in \mathscr {B}\left(\mathbb {R}^{m}\right)$, where $N = N(\varepsilon )$, such that for each $B \in \mathscr {M}$ there exist $S_{i}^{(\varepsilon )}$ and $S_{j}^{(\varepsilon )}$ with the properties $S_{i}^{(\varepsilon )} \subset B \subset S_{j}^{(\varepsilon )}$ and $\mathbb {Q}\left(S_{j}^{(\varepsilon )} \backslash S_{i}^{(\varepsilon )}\right) < \varepsilon$.

Show that the statement contained in Exercise 14 may become false if the condition that $\mathbb {Q}$ be absolutely continuous with respect to Lebesgue measure is dropped (for $m \geqslant 2$ ).

Prove that property (53) holds if $\mathscr {A}$ is the class of rectangles of the form $\left[a_{1}, b_{1}\right] \times \cdots \times \left[a_{d}, b_{d}\right] \subset [0,1]^{d}$.

Let $\mathscr {A} \subset \mathscr {B}\left([0,1]^{d}\right), d \geqslant 1$. For a function $F: \mathscr {A} \rightarrow \mathbb {R}$ denote

Set $F_{\delta }(A) = \operatorname {mes}\left(A^{(\delta )}\right), A \in \mathscr {A}, \delta > 0$ and $A^{(\delta )} = \left\{ x \in \mathbb {R}^{d}: \rho (x, \partial A) < \delta \right\}$, $\rho (x, \partial A) = \inf_{y \in \partial A} \rho (x, y), \rho$ is the Euclidean metric and mes is Lebesgue measure in $\mathbb {R}^{d}$.

Property (53) of a class of sets $\mathscr {A} \subset \mathscr {B}\left([0,1]^{d}\right)$ is the relation

Example 2. For a sequence $\xi_{1}, \xi_{2}, \ldots$ of independent identically distributed nondegenerate nonnegative random variables, the renewal process is defined by the formula

Throughout, the sum over the empty set is taken to be zero, so $X_{t}(\omega ) = 0$ if $\xi_{1}(\omega ) > t$.

Define the renewal function $u(t), t > 0$, as the mean number of renewals in the interval $(0, t]$, i.e., $u(t) = \mathbb {E}\left[X_{t}\right]$.

Prove that $u(t) < \infty$ for all $t > 0$.

Explain why the function $Z(B, \cdot )$ defined according to (14) is, for each $B \in \mathscr {B}(\mathrm{S})$, a random variable with values in the space $\left(\mathbb {Z}_{+}, \mathscr {A}\right)$, where $\mathbb {Z}_{+} = \left\{ 0,1, \ldots \right\} = \left\{ 0\right\} \cup \mathbb {N}, \mathscr {A}$ is the $\sigma$-algebra of all subsets of $\mathbb {Z}_{+}$. (It follows that $Z = \left\{ Z(B), B \in \mathscr {B}(\mathrm{S})\right\}$ is a real-valued stochastic process indexed by the sets $\mathscr {B}(\mathrm{S})$.)

Example 6. Let $\lambda$ be a finite measure ($\lambda \not\equiv 0$) defined on some measurable space $(\mathrm{S}, \mathscr {B})$. Let $Y, X_{1}, X_{2}, \ldots$ be independent random elements on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $Y$ is a real Poisson random variable with parameter $\lambda (\mathrm{S})$, and $X_{1}, X_{2}, \ldots$ are independent identically distributed $\mathscr {F} \mid \mathscr {B}$-measurable variables for which $\mathbb {P}_{X_{1}}\left(B\right) = \lambda (B) / \lambda (\mathrm{S})$ for $B \in \mathscr {B}$. Define on $\mathscr {B} \times \Omega$ the function $Z(B, \omega )$ by setting

($Z(B, \omega ) = 0$ when $Y(\omega ) = 0$). Formula (14) defines the so-called Poisson random measure with intensity measure (or leading measure) $\lambda$.

Example 6. Let $\lambda$ be a finite measure ($\lambda \not\equiv 0$) defined on some measurable space $(\mathrm{S}, \mathscr {B})$. Let $Y, X_{1}, X_{2}, \ldots$ be independent random elements on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $Y$ is a real Poisson random variable with parameter $\lambda (\mathrm{S})$, and $X_{1}, X_{2}, \ldots$ are independent identically distributed $\mathscr {F} \mid \mathscr {B}$-measurable variables for which $\mathbb {P}_{X_{1}}\left(B\right) = \lambda (B) / \lambda (\mathrm{S})$ for $B \in \mathscr {B}$. Define on $\mathscr {B} \times \Omega$ the function $Z(B, \omega )$ by setting

($Z(B, \omega ) = 0$ when $Y(\omega ) = 0$). Formula (14) defines the so-called Poisson random measure with intensity measure (or leading measure) $\lambda$.

Let $\lambda$ be a $\sigma$-finite measure on some measurable space ( $\mathrm{S}, \mathscr {B}$ ) $\left(\mathrm{S} = \bigcup_{m = 1}^{\infty } \mathrm{S}_{m}\right.$, where $\mathrm{S}_{m} \in \mathscr {B}, \mathrm{S}_{m} \cap \mathrm{S}_{k} = \varnothing$ for $m \neq k$ and $0 < \lambda \left(\mathrm{S}_{m}\right) < \infty$, $m, k \in \mathbb {N}$ ). In this case the construction of the Poisson random measure (14) is easily modified as follows.

On some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ construct a countable collection of independent variables

such that $Y^{(m)}$ has the Poisson distribution with parameter $\lambda_{m} = \lambda \left(\mathrm{S}_{m}\right)$ and $\operatorname {Law}\left(X_{j}^{(m)}\right) = \lambda (\cdot ) / \lambda \left(\mathrm{S}_{m}\right)$ on $\mathscr {B}_{m} = \mathscr {B} \cap \mathrm{S}_{m} ; m, j \in \mathbb {N}$. Let $Z_{m}(B, \omega )$ for $B \in \mathscr {B}_{m}$ be defined according to (14) with $Y$ replaced by $Y^{(m)}$ and $X_{j}$ by $X_{j}^{(m)}$. Set

Note that $Z(B, \omega )$ may also take infinite values.

Prove that if $\lambda (B) = \infty$ for some $B \in \mathscr {B}$, then $Z(B) = \infty$ a.s., and if $\lambda (B) < \infty$, then $Z(B) < \infty$ a.s.

Example 6. Let $\lambda$ be a finite measure ($\lambda \not\equiv 0$) defined on some measurable space $(\mathrm{S}, \mathscr {B})$. Let $Y, X_{1}, X_{2}, \ldots$ be independent random elements on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $Y$ is a real Poisson random variable with parameter $\lambda (\mathrm{S})$, and $X_{1}, X_{2}, \ldots$ are independent identically distributed $\mathscr {F} \mid \mathscr {B}$-measurable variables for which $\mathbb {P}_{X_{1}}\left(B\right) = \lambda (B) / \lambda (\mathrm{S})$ for $B \in \mathscr {B}$. Define on $\mathscr {B} \times \Omega$ the function $Z(B, \omega )$ by setting

($Z(B, \omega ) = 0$ when $Y(\omega ) = 0$). Formula (14) defines the so-called Poisson random measure with intensity measure (or leading measure) $\lambda$.

Let $\lambda$ be a $\sigma$-finite measure on some measurable space ( $\mathrm{S}, \mathscr {B}$ ) $\left(\mathrm{S} = \bigcup_{m = 1}^{\infty } \mathrm{S}_{m}\right.$, where $\mathrm{S}_{m} \in \mathscr {B}, \mathrm{S}_{m} \cap \mathrm{S}_{k} = \varnothing$ for $m \neq k$ and $0 < \lambda \left(\mathrm{S}_{m}\right) < \infty$, $m, k \in \mathbb {N}$ ). In this case the construction of the Poisson random measure (14) is easily modified as follows.

On some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ construct a countable collection of independent variables

such that $Y^{(m)}$ has the Poisson distribution with parameter $\lambda_{m} = \lambda \left(\mathrm{S}_{m}\right)$ and $\operatorname {Law}\left(X_{j}^{(m)}\right) = \lambda (\cdot ) / \lambda \left(\mathrm{S}_{m}\right)$ on $\mathscr {B}_{m} = \mathscr {B} \cap \mathrm{S}_{m} ; m, j \in \mathbb {N}$. Let $Z_{m}(B, \omega )$ for $B \in \mathscr {B}_{m}$ be defined according to (14) with $Y$ replaced by $Y^{(m)}$ and $X_{j}$ by $X_{j}^{(m)}$. Set

Note that $Z(B, \omega )$ may also take infinite values.

(see [32; p. 370]). Let $B \in \mathscr {B}$ be such that $0 < \lambda (B) < \infty$, where $\lambda$ is a $\sigma$-finite measure on $(\mathrm{S}, \mathscr {B})$. Let $B = \bigcup_{q = 1}^{n} B_{q}$, where $B_{q} \in \mathscr {B}, B_{j} \cap B_{q} = \varnothing$ for $q \neq j(q, j = 1, \ldots , n)$. Prove that for any $k_{1}, \ldots , k_{n} \in \mathbb {Z}_{+}$ and $k = \sum_{q = 1}^{n} k_{q}$ the following equality holds

Let $(\mathrm{S}_{t}, \mathscr {B}_{t})_{t \in T}$ be a family of measurable spaces, and let $\mathrm{S}_{T} = \prod_{t \in T} \mathrm{S}_{t}$ be their Cartesian product, consisting of the functions $y = y(t)$ defined on $T$ and such that $y(t) \in \mathrm{S}_{t}$ for $t \in T$.

Definition 6. The $\sigma$-algebra $\mathscr {B}_{T}$ in the space $\mathrm{S}_{T}$ generated by the collection of elementary cylinders, i.e., sets of the form $C_{T}(t, B_{t}) = \left\{ y \in \mathrm{S}_{T}: y(t) \in B_{t}\right\}$, where $B_{t} \in \mathscr {B}_{t}$, $t \in T$, is called the cylindrical $\sigma$-algebra.

Every random function $X = \left\{ X(t), t \in T\right\}$, regarded as a random element $\omega \mapsto X(\cdot , \omega )$ with values in $\mathrm{S}_{T}$, induces a probability distribution $\mathbb {P}_{X}$ on $(\mathrm{S}_{T}, \mathscr {B}_{T})$.

Give an example of real-valued processes $X = \left\{ X_{t}, t \in T\right\}$ and $Y = \left\{ Y_{t}, t \in T\right\}$ defined on the same probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $\mathbb {P}_{X} = \mathbb {P}_{Y}$ on $\mathscr {B}_{T}$, but $X$ and $Y$ are not (stochastically) equivalent. Give an example where $\mathbb {P}_{X} \neq \mathbb {P}_{Y}$, but the distributions of $X_{t}$ and $Y_{t}$ coincide for each $t \in T$.

Let $(\mathrm{S}_{t}, \mathscr {B}_{t})_{t \in T}$ be a family of measurable spaces, and let $\mathrm{S}_{T} = \prod_{t \in T} \mathrm{S}_{t}$ be their Cartesian product, consisting of the functions $y = y(t)$ defined on $T$ and such that $y(t) \in \mathrm{S}_{t}$ for $t \in T$.

Definition 6. The $\sigma$-algebra $\mathscr {B}_{T}$ in the space $\mathrm{S}_{T}$ generated by the collection of elementary cylinders, i.e., sets of the form $C_{T}(t, B_{t}) = \left\{ y \in \mathrm{S}_{T}: y(t) \in B_{t}\right\}$, where $B_{t} \in \mathscr {B}_{t}$, $t \in T$, is called the cylindrical $\sigma$-algebra.

Give an example of a (real-valued) process $X = \left\{ X_{t}, t \in T\right\}$ and sets $D, L \notin \mathscr {B}_{T}$ such that $\left\{ \omega : X(\cdot , \omega ) \in D\right\} \in \mathscr {F}$ and $\left\{ \omega : X(\cdot , \omega ) \in L\right\} \notin \mathscr {F}$.

Let $(\mathrm{S}_{t}, \mathscr {B}_{t})_{t \in T}$ be a family of measurable spaces, and let $\mathrm{S}_{T} = \prod_{t \in T} \mathrm{S}_{t}$ be their Cartesian product, consisting of the functions $y = y(t)$ defined on $T$ and such that $y(t) \in \mathrm{S}_{t}$ for $t \in T$.

Definition 6. The $\sigma$-algebra $\mathscr {B}_{T}$ in the space $\mathrm{S}_{T}$ generated by the collection of elementary cylinders, i.e., sets of the form $C_{T}(t, B_{t}) = \left\{ y \in \mathrm{S}_{T}: y(t) \in B_{t}\right\}$, where $B_{t} \in \mathscr {B}_{t}$, $t \in T$, is called the cylindrical $\sigma$-algebra.

Give an example of equivalent processes $X = \left\{ X_{t}, t \in T\right\}$ and $Y = \left\{ Y_{t}, t \in T\right\}$, and also of a set $D \notin \mathscr {B}_{T}$, such that $A = \left\{ \omega : X(\cdot , \omega ) \in D\right\} \in \mathscr {F}$ and $B = \left\{ \omega : Y(\cdot , \omega ) \in D\right\} \in \mathscr {F}$, but $\mathbb {P}\left(A\right) \neq \mathbb {P}\left(B\right)$.

Give an example of (stochastically) equivalent processes that are not indistinguishable.

Prove that if $\mathrm{S}$ is a separable metric space, then what stands under the probability sign in (56) is an event, i.e., a set belonging to $\mathscr {F}$.

Formula (56) is the condition of stochastic continuity on $T$ of a stochastic process $X = \left\{ X_{t}, t \in T\right\}$ defined on $(\Omega , \mathscr {F}, \mathbb {P})$, where $(T, \delta )$ is a pseudometric space, and taking, for each $t \in T$, values in a metric space $(\mathrm{S}, \rho )$: for each $t \in T$ and any $\varepsilon > 0$