Small sets for the Forward recurrence markov chain (a.k.a. residual lifetime chain)

Question

I came across the following result : consider a renewal process and the associated forward recurrence chain (a.k.a. residual lifetime chain), then every finite subset of integers is a small set.

I was wondering why this is true ?

Answer 1

I don't think it's true unless you add more assumptions.

Let $\{i_1, \ldots, i_d\}$ be a finite set of positive integers and let $V^+(n)$ be the forward recurrence chain.

If $i > 1$, $P(i,i-1) := P(V^+(n) = i-1 \mid V^+(n-1) = i) = 1$. On the other hand, $P(1,i) := P(V^+(n) = i \mid V^+(n-1) = 1) = p(i)$ where $p$ is the distribution for the interarrival times. I'll assume the support is some subset of the positive integers.

Notice that, for any $j$ in the finite set, and any $k$ in the state space: \begin{align*} P^{m^*}(j,k) &\ge P^{j-1}(j,1)p(k) P^{m^*-j}(k,k) \\ &= p(k) P^{m^*-j}(k,k) \end{align*} where $\max(i_1, \ldots, i_d) := m^*$.

If we could find some $\delta > 0$ such that make $P^{m^*-j}(k,k) > \delta$, then we would have our result. However, the last factor is the problem. It can easily be shown to be zero in certain circumstances (e.g. take $m^*-j=1$ and $k > 1$).