##
Zhiqiang Gao

Exact convergence rates in central limit theorems for a
branching random walk

### Résumé

Chen [Ann. Appl. Probab. 11 (2001), 1242--1262] derived exact convergence rates
in a central limit theorem and a local limit theorem for a supercritical branching
Wiener process. We extend Chen's results to a branching random walk under weaker
moment conditions. For the branching Wiener process, our results sharpen Chen's
by relaxing the second moment condition used by Chen to a moment condition of the
form $\E X |\ln X |^{1+\lambda}< \infty$.
In the rate functions that we find for a branching random walk,
we figure out some new terms which didn't appear in Chen's work.
The results are established in the more general framework, i.e. for a branching
random walk with a random environment in time. The lack of the second moment
condition for the offspring distribution and the fact that the exponential moment
does not exist necessarily for the displacements make the proof delicate;
the difficulty is overcome by a careful analysis of martingale convergence
using a truncating argument. The analysis is significantly more awkward
due to the appearance of the random environment.