Question

Let $a > 0$ and consider the Markov chain with state space N and transition matrix given by $$p_{i, i-1} = \frac{1}{a+1}; \quad p_{i, i+1} = \frac{a}{a+1}, \quad i \ge 1$$ and a reflecting barrier at state 0, such that $p_{0, 1} = 1$. (a) Show that when $a < 1$ this chain admits a stationary distribution of the form $$\pi_k = a^{k-1} \frac{1 - a^2}{2}, \quad k \ge 1$$ where the value of $\pi_0$ has to be determined. (b) Does the chain admit a stationary distribution when $a \ge 1$? (c) Show that the chain is positive recurrent when $a < 1$.

Name: let a 0 and consider the markov chain with state space n and transition matrix given by pi i 1 frac1a1 quad pi i1 fracaa1 quad i ge 1 and a reflecting barrier at state 0 such that p0 1 1 96183
Uploaded: 2024-04-02T02:57:32-08:00
Duration: 1 min 39 s
Channel: Dominador Tan
Description: let a 0 and consider the markov chain with state space n and transition matrix given by pi i 1 frac1a1 quad pi i1 fracaa1 quad i ge 1 and a reflecting barrier at state 0 such that p0 1 1 96183

          Let $a > 0$ and consider the Markov chain with state space N and transition matrix given by
$$p_{i, i-1} = \frac{1}{a+1}; \quad p_{i, i+1} = \frac{a}{a+1}, \quad i \ge 1$$
and a reflecting barrier at state 0, such that $p_{0, 1} = 1$.
(a) Show that when $a < 1$ this chain admits a stationary distribution of the form
$$\pi_k = a^{k-1} \frac{1 - a^2}{2}, \quad k \ge 1$$
where the value of $\pi_0$ has to be determined.
(b) Does the chain admit a stationary distribution when $a \ge 1$?
(c) Show that the chain is positive recurrent when $a < 1$.

$let a 0 and consider the markov chain with state space n and transition matrix given by pi i 1 frac1a1 quad pi i1 fracaa1 quad i ge 1 and a reflecting barrier at state 0 such that p0 1 1 96183$

Added by Kathleen T.

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