Let $M$ be a Markov chain, with (countable) state space $S$. (As will quickly become apparent, I'm picturing continuous time here, but I think everything I'm about to say is true in discrete time as long as the chain is aperiodic, and answers that only talk about discrete time are welcome.) For simplity, assume throughout that $M$ is not explosive. For $x\in S$, let $$T_x=\inf \{ t: M_t=x\text{ but }M_s\ne x\text{ for some }s<t \}$$ be the first time that the chain arrives at $x$ (or returns to $x$ if started from $x$). We say that $x\in S$ is transient if $\mathbb P_x(T_x<\infty)<1$, recurrent if $\mathbb P_x(T_x<\infty)=1$, and positive recurrent if $\mathbb E_x[T_x]<\infty$. Recurrent states which are not positive recurrent are null recurrent.
In my mind, the distinction between positive recurrent and null recurrent is important because of the relationship with limiting distributions: One can show that $x$ is a recurrent state, then the function $\pi(y):=\lim\limits_{t\to\infty}\mathbb P_x(M_t=y)$ is constant zero if $x$ is null recurrent and is a probability distribution if $x$ is positive recurrent.
But what if $x$ is a transient state? Here we don't quite get the same dichotomy, since if $M$ has absorbing states and the absorption probability when started from $x$ is strictly between $0$ and $1$, then $\pi$ is not constant zero but also not a probability distribution. However, I think the distinction between whether $\pi$ is a probability distribution or not is interesting.
So I propose the following (provisional) definition: A transient state $x$ is null transient if $\sum_{y\in S}\lim\limits_{t\to\infty}\mathbb P_x(M_t=y)<1$.
Question: Is this a real thing? What is it actually called?
Additional thoughts:
- I feel compelled to point out that this distinction is meaningful. If $S$ is finite, no $x$ can be null transient. But, for example, if $M$ has a single absorbing state $a$ and the absorption probability $\mathbb P_x(T_a<\infty)$ from $x$ is less than $1$, then $x$ is null transient.
- Put another way, I'm asking if there's a name for the property that $x$ is transient but the process when started in state $x$ converges to a limiting distribution.