Rate of convergence for polymers in a weak disorder

We consider directed polymers in random environment on the lattice Z d at small inverse temperature and dimension d $\ge$ 3. Then, the normalized partition function W n is a regular martingale with limit W. We prove that n (d--2)/4 (W n -- W)/W n converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale W n are different from those for polymers on trees

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

Chen [Ann. Appl. Probab. {\bf 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.Comment: Corrected version of https://hal.archives-ouvertes.fr/hal-01095105. arXiv admin note: text overlap with arXiv:1504.01181 by other author

Convergence in LpL^p and its exponential rate for a branching process in a random environment

We consider a supercritical branching process $(Z_n)$ in a random environment $\xi$. Let $W$ be the limit of the normalized population size $W_n=Z_n/E[Z_n|\xi]$. We first show a necessary and sufficient condition for the quenched $L^p$ ($p>1$) convergence of $(W_n)$, which completes the known result for the annealed $L^p$ convergence. We then show that the convergence rate is exponential, and we find the maximal value of $\rho>1$ such that $\rho^n(W-W_n)\rightarrow 0$ in $L^p$, in both quenched and annealed sense. Similar results are also shown for a branching process in a varying environment

Limit theorems for decomposable branching processes in a random environment

We study the asymptotics of the survival probability for the critical and decomposable branching processes in random environment and prove Yaglom type limit theorems for these processes. It is shown that such processes possess some properties having no analogues for the decomposable branching processes in constant environmentComment: 21 page

Moments, moderate and large deviations for a branching process in a random environment

Let $(Z_{n})$ be a supercritical branching process in a random environment $\xi$, and $W$ be the limit of the normalized population size $Z_{n}/\mathbb{E}[Z_{n}|\xi ]$. We show large and moderate deviation principles for the sequence $\log Z_{n}$ (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of $W$, and show an equivalence for all the moments of $Z_{n}$. Central limit theorems on $W-W_n$ and $\log Z_n$ are also established