$\sum_{j} p_{ij}$ For any Markov chain, what is the value of ? (the sum of all $p_{ij}$ across a row) 0 1 $p_i$ cannot be generalized
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Step 1: A Markov chain is a stochastic process that satisfies the Markov property, which states that the conditional probability distribution of future states of the process depends only upon the present state, not on the sequence of events that preceded it. Show more…
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Recall that state $i$ is said to be positive recurrent if $m_{i, i}<\infty$, where $m_{i, i}$ is the expected number of transitions until the Markov chain, starting in state $i$, makes a transition back into that state. Because $\pi_{i}$, the long run proportion of time the Markov chain, starting in state $i$, spends in state $i$, satisfies $$ \pi_{i}=\frac{1}{m_{i, i}} $$ it follows that state $i$ is positive recurrent if and only if $\pi_{i}>0$. Suppose that state $i$ is positive recurrent and that state $i$ communicates with state $j .$ Show that state $j$ is also positive recurrent by arguing that there is an integer $n$ such that $$ \pi_{j} \geqslant \pi_{i} P_{i, j}^{n}>0 $$
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Consider a Markov chain with state space 0, 1, 2, ... and transition probabilities p_{i,i-1} = 1, i = 1, 2, 3, ... p_{0,i} = p_i, i = 0, 1, 2, 3, ... where p_i > 0 for all i and sum_{i >= 0} p_i = 1. (c) What is the period of state i, for all values of i? (d) Under what condition is the chain positive recurrent? (e) If the chain is positive recurrent, what is the mean number of steps required for it to return to state i if it starts from i?
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