# Degeneration, probability of

The probability of no particles being left in a branching process at an epoch $t$. Let $\mu ( t)$ be the number of particles in a branching process with one type of particles. The probability of degeneration

$${\mathsf P} _ {0} ( t) = \ {\mathsf P} \{ \mu ( t) = 0 \mid \mu ( 0) = 1 \}$$

does not decrease as $t$ increases; the value

$$q = \lim\limits _ {t \rightarrow \infty } {\mathsf P} _ {0} ( t)$$

is called the probability of degeneration in infinite time or simply the probability of degeneration. If $\tau$ is the time elapsing from the beginning of the process to the epoch of disappearance of the last particle, then ${\mathsf P} \{ \tau < t \} = {\mathsf P} _ {0} ( t)$ and ${\mathsf P} \{ \tau < \infty \} = q$. The rate of convergence of ${\mathsf P} _ {0} ( t)$ to $q$ as $t \rightarrow \infty$ has been studied for various models of branching processes.