Some limit theorems for the critical Galton-Watson branching processes (Q6083225)

Let \((\xi_{k,j})_{k,j \in \mathbb{N}}\) be a family of i.\,i.\,d.\ nonnegative integer-valued random variables independent of the nonnegative integer-valued random variable \(\eta\). Write \(\xi\) for a copy of \(\xi_{1,1}\) and \(p_k = \mathbb{P}(\xi=k)\), \(k=0,1,2,\ldots\) Now consider the Galton-Watson branching process \((W_n)_{n \in \mathbb{N}_0}\) with initial population number \(W_0 = \eta\) and the usual recursive definition \[ W_n = \sum_{j=1}^{W_{n-1}} \xi_{n,j}, \quad n \in \mathbb{N}. \] The Galton-Watson process \((W_n)_{n \in \mathbb{N}_0}\) is assumed non-degenerate and critical, i.e., \(p_0+p_1 \not = 1\) and \(\mathbb{E}[\xi]=1\), respectively. Let \(F\) denote the generating function of \(\xi\), that is, \(F(s) = \mathbb{E}[s^\xi]\), \(0 \leq s \leq 1\). Consider Slack's condition for the generating function \[ \hspace{-3.5cm} \text{(S)} \qquad\qquad\qquad\qquad\qquad F(s) = s + (1-s)^{1+\alpha} L(1-s) \] where \(\alpha \in (0,1]\) and \(L\) is slowly varying at \(0\). If additionally, \(\mathbb{E}[\eta^2]<\infty\), the authors find the asymptotic of the survival probability \(\mathbb{P}(W_n > 0)\) as \(n \to \infty\). Further, the authors provide limit theorems for the distribution of \(W_n\) conditional given \(W_n > 0\) as \(n \to \infty\) making different assumptions on the law of \(\eta\). \begin{itemize} \item Assuming \(\mathbb{E}[\eta^2]<\infty\); the resulting limit theorem generalizes a theorem of Slack. \item Assuming that \(\eta=\eta_n\) is a deterministic sequence tending to \(\infty\) as \(n \to \infty\) at a particular rate. This is in the spirit of classical work by \textit{W. Feller} [Diffusion processes in genetics. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, pp.~227--246 (1951; Zbl 0045.09302)]. \item Assuming \(h(s) = \mathbb{E}[s^\eta] = 1-(1-s)^\theta L_0(\frac1{1-s})\) as \(s \to 1\) with \(\theta \in (0,1)\). In particular, \(\mathbb{E}[\eta]=\infty\) in this case. \end{itemize} The proofs are based on computations involving Laplace transforms and generating functions.