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.

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The probability of degeneration is more commonly called the probability of extinction (in infinite time).

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