Math Problems Vault

Problem 8.2.1

(a)(b)

Let $\left\{ X_{n}\right\}$ be a Markov chain on the state space $S = \left\{ 0,1\right\}$ with transition matrix

\mathbf{P} =\left[\begin{smallmatrix} 1-\alpha & \alpha \\ \beta & 1-\beta \end{smallmatrix}\right]

where $\alpha +\beta > 0$ . Find:

?

(a)

the correlation $\rho \left(X_{m}, X_{m+n}\right)$ , and its limit as $m \rightarrow \infty$ with $n$ remaining fixed,

(b)

$\lim_{n \rightarrow \infty } n^{-1} \sum_{r = 1}^{n} \mathbb {P}\left(X_{r} = 1\right)$ , and the condition under which the process is strongly stationary.

Problem 8.2.2

(a)(b)

Let $\left\{ N(t): t \geq 0\right\}$ be a Poisson process of intensity $\lambda$ , and let $T_{0}$ be an independent random variable such that $\mathbb {P}\left(T_{0} = \pm 1\right) = \frac{1}{2}$ . Define $T(t) = T_{0}(-1)^{N(t)}$ . Show that $\left\{ T(t): t \geq 0\right\}$ is stationary and find:

?

(a)

$\rho (T(s), T(s+t))$ ,

(b)

the mean and variance of $X(t) = \int_{0}^{t} T(s) d s$ . [The so-called Goldstein-Kac process $X(t)$ denotes the position of a particle moving with unit speed, starting from the origin along the positive $x$ -axis, whose direction is reversed at the instants of a Poisson process.]

Problem 8.2.3

An integer-valued counting process $\left\{ N(t): t \geq 0\right\}$ with $N(0) =$ 0 is called crudely stationary if $p_{k}(s, t) = \mathbb {P}\left(N(s+t)-N(s) = k\right)$ depends only on the length $t-s$ and not on the location $s$ . It is called simple if, almost surely, it has jump discontinuities of size 1 only. Show that, for a simple crudely stationary process $N, \lim_{t \downarrow 0} t^{-1} \mathbb {P}\left(N(t) > 0\right) = \mathbb {E}\left[N(1)\right]$ .

?

Problem 8.2.4

Let $X = \left\{ X_{n}: n \geq 0\right\}$ be an ergodic Markov chain started in its stationary distribution $\pi$ . Let $A$ be a subset of states, with stationary probability $\pi (A)$ , and let $T_{A} = \min \left\{ n \geq 1: X_{n} \in A\right\}$ . Show that $\mathbb {E}\left[T_{A} \mid X_{0} \in A\right] = 1 / \pi (A)$ .

?