Large Deviation Rates for Branching Processes--I. Single Type Case

Abstract

Let $\{Z_n\}^\infty_0$ be a Galton-Watson branching process with offspring distribution $\{p_j\}^\infty_0$. We assume throughout that $p_0 = 0, p_j \neq 1$ for any $j \geq 1$ and $1 < m = \Sigma jp_j < \infty$. Let $W_n = Z_nm^{-m}$ and $W = \lim_nW_n$. In this paper we study the rates of convergence to zero as $n \rightarrow \infty$ of $P\big(\big|\frac{Z_{n+1}}{Z_n} - m\big| > \varepsilon\big),\quad P(|W_n - W| > \varepsilon)$, $P\big(\big|\frac{Z_{n+1}}{Z_n} - m \mid > \varepsilon\big|W \geq a\big)$ for $\varepsilon > 0$ and $a > 0$ under various moment conditions on $\{p_j\}$. It is shown that the rate for the first one is geometric if $p_1 > 0$ and supergeometric if $p_1 = 0$, while the rates for the other two are always supergeometric under a finite moment generating function hypothesis.