From law of the iterated logarithm to Zolotarev distance for supercritical branching processes in random environment
Yinna Ye
TL;DR
This work analyzes supercritical branching processes in an i.i.d. random environment and establishes a law of the iterated logarithm, a strong law, an invariance principle, and optimal convergence rates in the central limit theorem for the log-population $\log Z_n$, all under Zolotarev and Wasserstein distances. The key idea is the decomposition $\log Z_n = S_n + \log W_n$ with $S_n = \sum_{i=0}^{n-1} X_i$ and $W_n = Z_n/\Pi_n$, enabling martingale limit techniques to control fluctuations and obtain precise distance-based convergence rates. Under suitable moment and compatibility conditions (Con1, Con2), the paper derives $\zeta_r$ and $W_r$ bounds to quantify convergence to the normal law, and proves an LIL and an invariance principle that relate the growth of $\log Z_n$ to a Brownian motion with variance $\sigma^2$. These results advance the understanding of distributional limits for BPREs and provide sharp metrics for convergence useful in applications and further theoretical development.
Abstract
Consider $(Z_n)_{n\geq0}$ a supercritical branching process in an independent and identically distributed environment. Based on some recent development in martingale limit theory, we established law of the iterated logarithm, strong law of large numbers, invariance principle and optimal convergence rate in the central limit theorem under Zolotarev and Wasserstein distances of order $p\in(0,2]$ for the process $(\log Z_n)_{n\geq0}$.