From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment (Q6606007)

The author considers a supercritical branching process in a random environment \((Z_n)_{n\geq0}\) given by \(Z_0=1\) and \(Z_{n+1}=\sum_{i=1}^nX_{n,i}\) for \(n\geq0\), where there exist IID random variables \(\xi_0,\xi_1,\ldots\) such that, conditional on \(\xi_n\), we have that \((X_{n,i})_{i\geq1}\) is a sequence of IID random variables; these latter random variables also independent of \(Z_1,\ldots,Z_n\). Within this framework, the author uses martingale limit theory to establish limiting properties of \((\log Z_n)_{n\geq0}\), including a law of the iterated logarithm, a strong law of large numbers, an invariance principle, and central limit theorems with explicit rates of convergence to a Gaussian limit in the Zolotarev and Wasserstein distances of order \(p\in(0,2]\).