# Positive recurrence of a continuous-time jump process, from its jump chain

I am looking at an irreducible, continuous-time jump process $(X_t)_{t\geq0}$ with the following jump times. Let a Poisson process $(T_i)_{i=1}^\infty$ determine the event times. With probability $0<p<1/2$, the chain does nothing. Else, with probability $1-p$, it moves somewhere giving a jump time $J_i$. Thus $T_i\leq J_i$ for all $i$.

I know that the jump chain $(X_{J_i})_{i=1}^\infty$ is ergodic, from other parts of my work. How can I conclude that the original chain is ergodic? My intuition is that jumps still occur sufficiently frequently (because with high probability, there are at least $t/2$ events by time $t$).

In this case it is particularly easy because the jump time are independent of where you go. This is so because they are the same exponentials determined by the poisson rate & the coin flip in every state. The return time to $0$ can be written as $\sum ^N T^*_i$ where N is the return index of the jump chain and the $T^*_i$ are $poi((1-p)\lambda)$ independent of N. You probably know how to handle this sort of compound distribution. – mike May 9 '12 at 16:26