Jump chain

The jump chain records the successive states visited by a continuous-time Markov chain while discarding its holding times.

On $\mathbb{N}$, let the chain jump from every $i\ge 1$ to $i+1$ with probability $p$ and to $i-1$ with probability $1-p$, and jump from zero to one. It is recurrent exactly when $p\le 1/2$; at $p=1/2$ it is null recurrent.

For holding rates $q_i$, a measure $\pi$ satisfies $\pi Q=0$ exactly when ${\mu }_i=q_i{\pi }_i$ is invariant for the jump chain. Thus ${\pi }_i={\mu }_i/q_i$ when normalization is possible.

Even when the jump chain has invariant probability $\mu$, the continuous-time invariant measure ${\mu }_i/q_i$ may have infinite mass if holding rates become too small, producing infinite mean return time.

A recurrent jump chain with infinite invariant measure $\mu$ can yield a positive-recurrent continuous-time chain when ${\sum }_i{\mu }_i/q_i<\infty$.

If the jump chain visits a state of finite holding rate infinitely often, the sum of its independent positive holding times at that state diverges almost surely, preventing explosion.

For a continuous-time Markov chain whose jump times tend to infinity, the chain and its jump chain have the same directed paths of positive-probability jumps. They are therefore irreducible simultaneously, and a state is recurrent for one exactly when it is recurrent for the other: the event of returning after departure depends only on the sequence of visited states.