Some convergence theorems on a supercritical Galton-Watson process (Q1069569)

Let \(X=(X_ n,n\geq 0;X_ 0=1)\) be a supercritical Galton-Watson process with offspring mean m \((1<m<\infty)\) and variance \(\sigma^ 2\) \((0<\sigma^ 2<\infty)\). Let \(\hat m\) be the estimator of m given by \(\hat m=\sum^{N}_{n=0}X_{n+1}/\sum^{N}_{n=0}X_ n.\) The limiting distribution of \[ (\sum^{N}_{n=0}X_ n)^{- 1/2}(\sum^{N}_{n=0}X_{n+r}-\hat m^ r\sum^{N}_{n=0}X_ n),\quad r\geq 2, \] is derived. As an application of this result, some limit theorems leading ultimately to a parameter free result of statistical interest are also established.