Classification of states of continuous-time chains

Let $q_{i,i+1} = 2^{i-1}$ for all $i \geq 0$ and let $q_{i,i-1} = 2^{i-1}$ for all $i \geq 1$. All other transition rates are $0$.

Let $X = (X(t): t \geq 0)$ be a Markov chain on $\mathbb {Z}$ with generator $\mathbf{G} = \left(g_{i j}\right)$ given by

Show that $X$ is recurrent. Is $X$ positive recurrent?

Let $X$ be a Markov chain on $\mathbb {Z}$ with generator $G = \left(g_{i j}\right)$ satisfying

Show that $X$ is transient, but has an invariant distribution. Explain.

Let $X$ be a continuous-time Markov chain with generator $\mathbf{G} = \left(g_{i j}\right)$. Show that $X$ is irreducible if and only if for any pair $i, j$ of distinct states there exists a sequence $i, k_{1}, k_{2}, \ldots , k_{n}, j$ of distinct states such that $g_{i, k_{1}} g_{k_{1}, k_{2}} \cdots g_{k_{n}, j} > 0$.

Let $X$ be a continuous-time Markov chain with transition probabilities $p_{i j}(t)$ and define $F_{i} = \inf \left\{ t > T_{1}: X(t) = i\right\}$ where $T_{1}$ is the time of the first jump of $X$. Show that, if $g_{i i} \neq 0$, then $\mathbb {P}_{i}\left(F_{i} < \infty \right) = 1$ if and only if $i$ is recurrent.

Let $X$ be the simple symmetric random walk on the integers in continuous time, so that

Show that the walk is recurrent. Let $T$ be the time spent visiting $m$ during an excursion from 0. Find the distribution of $T$.

Let $i$ be a transient state of a continuous-time Markov chain $X$ with $X(0) = i$. Show that the total time spent in state $i$ has an exponential distribution.

Let $X$ be an asymmetric simple random walk in continuous time on the non-negative integers with retention at $0$, so that

Suppose that $X(0) = 0$ and $\lambda > \mu$. Show that the total time $V_{r}$ spent in state $r$ is exponentially distributed with parameter $\lambda -\mu$.

Assume now that $X(0)$ has some general distribution with probability generating function $G$. Find the expected amount of time spent at 0 in terms of $G$.

Let $Z_{n} = X(n h)$ where $X$ is a Markov chain and $h > 0$. Show that $i$ is recurrent for $Z$ if and only if it is recurrent for $X$. Show that $Z$ is irreducible if and only if $X$ is irreducible.

Prove that, for the embedded Markov chain $\xi_{n}^{*}$, the states $x_{i}$ and $x_{j}$ communicate if and only if $p_{ij}(t) > 0$ for all $t > 0$.

Let a jump Markov process with values 1, 2, 3, 4, and 5 be defined by the transition intensity matrix

Find the transition probability matrix and classify the states.

A homogeneous jump Markov process with values 1, 2, 3, 4, 5, and 6 is given by the transition intensity matrix

A homogeneous jump Markov process $\xi (t)$ with values 1, 2, 3, 4, 5, and 6 is given by the transition intensity matrix