Large Deviations for a Critical Galton-Watson Branching Process

Abstract

In this paper, a critical Galton-Watson branching process {Z_n} is considered. Large deviation rates of \({S_{{Z_n}}}: = \sum\limits_{i = 1}^{{Z_n}} {{X_i}} \) are obtained, where {X_i, i ≥ 1} is a sequence of independent and identically distributed random variables and X₁ is in the domain of attraction of an α-stable law with α ∈ (0, 2). One shall see that the convergence rate is determined by the tail index of X₁ and the variance of Z₁. Our results can be compared with those ones of the supercritical case.