Why does the limit defining the transition rate for a Markov chain exist?

Question

Background

Let $S$ be a countable (finite or infinite) state space. The text I am reading defines a Markov process as a stochastic process $\{X(t)\}_{t\in\mathbb R_{\ge0}}$ with the Markov property: For any times $t_1<\cdots<t_n<t_{n+1}$ and any $j_1,\cdots,j_{n+1}\in S$ such that $P(X(t_1)=j_1,\cdots,X(t_n)=j_n)>0$,

$\begin{align} P(X(t_{n+1})=j_{n+1}|X(t_1)=j_1,\cdots,X(t_n)=j_n)=P(X(t_{n+1})=j_{n+1}|X(t_n)=j_n). \end{align}$

The process is time homogeneous if $P(X(t+\tau)=k|X(t)=j)$ does not depend on $t$ and is irreducible if every state can be reached from any other state.

Transition Rate

Let $X$ an irreducible, time homogeneous Markov process which remains in each state for a positive length of time and cannot pass through an infinite number of states in finite time. The transition rate from state $j$ to $k$ is defined as

$\begin{align} q(j,k)=\lim_{\tau\to0}\frac{P(X(t+\tau)=k|X(t)=j)}\tau \end{align}$

for $j\ne k$ and $q(j,j)=0$.

The things that bug me about this definition are the two-sided limit (instead of $\tau\to0^+$), plus I don't see why the limit needs to exist.

My understanding is that the $q(j,k)$, when suitably normalized, should give the probability of transitioning from $j$ to $k$ the next time the process jumps. If that's right then I'm pretty confident that the limit $\tau\to0^-$, when normalized, should be the probability of having come from $k$ the last time the process jumped, given that it's currently at $j$. Which wouldn't be equal to the right-hand limit, in general. So I'm fairly confident that $\tau\to0^+$ is meant here.

More concerning is the existence of the limit. I assume it does exist as otherwise not mentioning that possibility would be a huge omission, but I don't yet see how to prove this.

Questions

Why must the limit defining $q(j,k)$ exist?

And (less importantly):

Am I right in thinking that the limit should be taken from the right only?

What I Tried

I Googled for information on transition rates, but it looks to me like most places define Markov processes in terms of the transition rates and prove that they satisfy the Markov property, instead of defining them in terms of the Markov property and proving that the transition rates are well-defined. So that wasn't much help.

Also, a search of this site ("transition rate" [markov-chains]) didn't turn up anything that looked relevant in the first couple of pages of results.

Answer 1

Today I stumbled across a text by Ward Whitt about continuous time Markov chains. On pages 7 and 8 he addresses this question. He doesn't prove that the limit exists (for him it is an assumption), but he does provide a citation to sections II.1 and II.2 of Kai Lai Chung's 1967 book Markov Chains with Stationary Transition Probabilities. I managed to find a copy of the first edition of Chung's book, and the existence of the limit is Theorem 5 in section II.2 there. I won't reproduce the proof here because it builds on some 13 pages of general theory of transition matrices, but Chung's book is easy enough to find online for those interested.