= Solution
An irreducible <Markov chain> has <positive recurrent states> if the expected first return time to a state is finite; this then holds for every state. A recurrent chain with infinite mean return time has <null recurrent states>.
An irreducible positive recurrent chain admits a <stationary distribution>. If such a distribution $\pi$ existed for this walk, its stationarity equations would give
$$
\pi_j=\tfrac12(\pi_{j-1}+\pi_{j+1})\quad(j\in\mathbb Z).
$$
Thus $\pi_{j+1}-\pi_j=\pi_j-\pi_{j-1}$, so $\pi_j=A+Bj$. Nonnegativity for all positive and negative integers forces $B=0$. A constant sequence cannot sum to one over $\mathbb Z$, so no stationary probability distribution exists. Hence \b[the walk is null recurrent, not positive recurrent]: all its states are <null recurrent states>. The first-return generating function in part (c) also verifies directly that its mean return time is infinite.
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