Question

4. Let {X_n}_{n=0}^infty be a Markov chain with state space {0, 1, 2, ...}. The transition probabilities are $p_{0,1} = 1$, $p_{i, i+1} + p_{i, i-1} = 1$, $p_{i, i+1} = left(frac{i+1}{i} ight)^2 p_{i, i-1}$, $i ge 1$ (i) Suppose that $X_0 = 0$. Find the probability that the chain never returns to state 0. (ii) Now suppose that the transition probabilities are $p_{0,1} = 1$, $p_{i, i+1} + p_{i, i-1} = 1$, $p_{i, i+1} = left(frac{i+1}{i} ight)^alpha p_{i, i-1}$, $i ge 1$ for some constant $alpha$. For every $alpha in (-infty, infty)$, indicate whether the chain is positive recurrent, null recurrent, or transient. Prove your conclusion.

          4. Let {X_n}_{n=0}^infty be a Markov chain with state space {0, 1, 2, ...}. The transition probabilities are
$p_{0,1} = 1$, $p_{i, i+1} + p_{i, i-1} = 1$, $p_{i, i+1} = left(frac{i+1}{i}
ight)^2 p_{i, i-1}$, $i ge 1$
(i) Suppose that $X_0 = 0$. Find the probability that the chain never returns to state 0.
(ii) Now suppose that the transition probabilities are
$p_{0,1} = 1$, $p_{i, i+1} + p_{i, i-1} = 1$, $p_{i, i+1} = left(frac{i+1}{i}
ight)^alpha p_{i, i-1}$, $i ge 1$
for some constant $alpha$. For every $alpha in (-infty, infty)$, indicate whether the chain is positive recurrent, null recurrent, or transient. Prove your conclusion.

$4. Let Xnn=0^infty be a Markov chain with state space 0, 1, 2, .... The transition probabilities are p0,1 = 1, pi, i+1 + pi, i-1 = 1, pi, i+1 = left(fraci+1i ight)^2 pi, i-1, i ge 1 (i) Suppose that X0 = 0. Find the probability that the chain never returns to state 0. (ii) Now suppose that the transition probabilities are p0,1 = 1, pi, i+1 + pi, i-1 = 1, pi, i+1 = left(fraci+1i ight)^alpha pi, i-1, i ge 1 for some constant alpha. For every alpha in (-infty, infty), indicate whether the chain is positive recurrent, null recurrent, or transient. Prove your conclusion.$

Added by Daniel O.

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