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 , 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 $t$). Thanks for your help, Derek
Positive recurrence of a continuous-time jump process, from its jump chain
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$\begingroup$
probability
stochastic-processes
markov-chains
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1In 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. – 2012-05-09
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0Hi Mike, thanks for pointing me in the right direction. Think I've got it! – 2012-05-09