Some Limit Theorems on a Supercritical Simple Galton-Watson Process

Abstract

Let $X = (X_n; n \geq 0; X_0 = 1)$ be a supercritical Galton-Watson process possessing an offspring mean $1 < m < \infty$, and variance $0 < \sigma^2 < \infty$. The limiting distribution of $\{X^{-1/2}_n(X_{n+r} - \hat{m}^rX_n); r = 2, \cdots, T\}$ where $\hat{m} = X_{n+1}/X_n$, is obtained. As a consequence of this result a Quenouille-Bartlett type of asymptotic goodness of fit test is also proposed for the process $X$.