Branching processes with negative offspring distributions
Let \(Z_n\), \(n=0,1,2,\dots\), be i.i.d. random variables, define \(Y_0:=1\), \(Y_i:= Y_{i-1}+ 1-Z_i\), \(i=1,2, \dots\), and let \(T\) be the minimal \(i\) for which \(Z\leq 0\), with \(T=\infty\) if no such \(i\) exists. The escape probability \(P(T= \infty)\) is computed first purely probabilistically and then again analytically, employing a method that has already been used successfully with branching processes. It also leads to the asymptotics of \(P(T>N)\) and \(P(T=\infty \mid T>N)\).
scientific article; zbMATH DE number 1964623