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For the simple symmetric random walk on $\mathbb{Z}$, the renewal equation relates first-return and return probability generating functions. The central-binomial return series is $U(s)=(1-s^2{)}^{-1/2}$, giving $F(s)=1-U(s{)}^{-1}$. Its value tends to one at $s=1$, but its derivative diverges there: the walk has null recurrent states.

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 giveThus ${\pi }_{j+1}-{\pi }_j={\pi }_j-{\pi }_{j-1}$, 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 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.

Let $u_n={\mathbb{P}}_0(X_n=0)$ and $f_n={\mathbb{P}}_0(T=n)$. Decomposing at the first return and using the Markov property gives the renewal equationFor $0<s<1$, define the probability generating functions $U(s)={\sum }_{n\ge 0}{u}_n{s}^n$ and $F(s)={\sum }_{n\ge 1}{f}_n{s}^n={\mathbb{E}}_0[s^T]$. Absolute convergence, or nonnegative summation, justifies the convolution identity $U=1+FU$. The binomial series givesSolving the renewal identity gives the first-return generating function of the simple symmetric random walkAs consistency checks, $F(s)\to 1$ as $s\uparrow 1$, confirming recurrence, while $F^{\prime }(s)=s/\sqrt{1-s^2}\to \infty$. By monotone convergence of ${\sum }_{n}n{f}_n{s}^{n-1}$, this last limit is ${\mathbb{E}}_{0}T=\infty$, confirming null recurrent states.

False. Take the chain from part (b) with $\lambda =1$. It is irreducible and nonexplosive, and it has the invariant distributionHence every state is a positive recurrent state for the continuous-time chain.

Its jump chain is the reflected simple symmetric random walk on $\mathbb{N}$. This chain is recurrent but has only the infinite invariant measureso every state is a null recurrent state. Equivalently, the invariant-measure transfer between a jump chain and a CTMC divides this infinite jump-chain measure by the rapidly growing holding rates and produces the summable continuous-time invariant measure. Thus fast holding rates create continuous-time positive recurrence, and positive recurrence need not pass from a continuous-time chain to its jump chain.