Consider a jump process \((X_{t})\) determined by transition matrix \(P=(p_{i,j})\) with \(p_{i,i}=0\), and holding rate function \((q_{k}=q(k),~k \in S)\), \(0<q_{k}<\infty\). Then jump times \(S_{0}=0, S_{1}, S_{2}, \dots\) are constructed via holding times \[ T_{m-1}=S_{m}-S_{m-1}= \frac{E_{m-1}}{q(X_{m-1})}\sim\mathcal{E}(q(X_{m-1})), \] where \(\{ E_{m} \}\) is a sequence of i.i.d. \(\mathcal{E}(1)\) random variables. At times \(S_{m}\), \(m=1,2,\dots\) the chain jumps to state \(X_{m}=j\) according to probabilities \(p_{i,j}\). Further, \(X_{t}=X_{m}\) on \(S_{m}\leq t\leq S_{m+1}\) so that the chain is well defined on \([0,S_{\infty})\) with \(S_{\infty}=\lim_{n \to \infty}S_{n}\).
We consider the following problems:
- When is the limit \(S_{\infty}=\infty\)? - i.e. when \((X_{t})\) is defined for all \(t \geq 0\).
- What is the important of the condition \(0<q_{k}<\infty\)?
State \(i\) for which \(0<q_{i}<\infty\) is called stable. If \(q_{i}=\infty\), state \(i\) is called instantaneous.
Note that: - \(q_{i}=\infty\) means that the mean holding time at state \(i\) is \(\frac{1}{q_{i}}=0\). Such state is exited instantaneously. - \(q_{i}=0\) means that the mean holding time at state \(i\) is \(\frac{1}{q_{i}}=\infty\). Thus \(i\) is an absorbing state.
Assumption: For the following results we assume all chains considered have only stable states.
If for all \(i \in S\) we have \(\mathbb{P}(S_{\infty}=\infty|X_{0}=i)=1\) then the chain \((X_{t})\) is called regular.
Note that if \(S_{\infty}<\infty\) this intuitively means that an infinite number of jumps at times \(S_{1}, S_{2}, \dots\) occurs on finite interval \([0,S_{\infty})\). This is called an explosion.
For all \(i \in S\) we have that \(\mathbb{P}(S_{\infty}<\infty|X_{0}=i)=\mathbb{P}\left( \sum_{n} \frac{1}{q(X_{n})}<\infty \middle| X_{0}=i \right).\) Thus we have that \[(X_{t})~\text{is regular}\iff \mathbb{P}\left( \sum_{n} \frac{1}{q(X_{n})}=\infty\middle|X_{0}=i \right)=1,~~\forall i \in S.\]
\begin{proof} First we note that \[
S_{\infty}=\lim_{N \to \infty}S_{N}=\lim_{N \to \infty}\sum_{n=0}^{N-1}(S_{n+1}-S_{n})=\sum_{n=0}^\infty(S_{n+1}-S_{n})=\sum_{n=0}^\infty \frac{{E_n}}{q(X_{n})}.
\] Then we have that \[
\begin{align}
\mathbb{P}(S_{\infty} < \infty|X_{0}=i_{0}=i, X_{n}=i_{n}, n \geq 0) & = \mathbb{P}\left( \sum_{n=0}^\infty \frac{E_{n}}{q(X_{n})}<\infty\middle|(X_{n}=i_{n},~n\geq 0) \right) \\
& = \mathbb{P}\left( \sum_{n=0}^\infty \frac{1}{q(X_{n})}<\infty\middle|(X_{n}) \right),
\end{align}
\] where this last equality follows from properties of the exponential distribution and the [[tower-property|tower property]]. \end{proof} Some examples of regular processes are:
- Poisson Process is regular since \(\sum_{i} \frac{1}{\lambda}=\infty\).
- [[yule-process]] is regular since \(\sum_{i} \frac{1}{i\lambda}=\infty\).
- [[pure-birth-process]] is regular if and only if \(\sum_{i}\frac{1}{\lambda_{i}}=\infty\).
The following are sufficient conditions for regularity: 1. \(\max_{i}q_{i}<\infty\) 2. Finite state space \(S\) 3. \(X_{0}=i\) and \(i\) is a persistent state for the imbedded jump chain \((X_{t})\).