# Iksanov O. M.

### Some moment results about the limit of a martingale related to the supercritical branching random walk and perpetuities

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 451–471

Let $\mathcal{M}_{(n)},\quad n = 1, 2,...,$ be the supercritical branching random walk in which the family sizes may be infinite with positive probability. Assume that a natural martingale related to $\mathcal{M}_{(n)},$ converges almost surely and in the mean to a random variable $W$. For a large subclass of nonnegative and concave functions $f$ , we provide a criterion for the finiteness of $\mathbb{E}W f(W)$. The main assertions of the present paper generalize some results obtained recently in Kuhlbusch’s Ph.D. thesis as well as previously known results for the Galton-Watson processes. In the process of the proof, we study the existence of the $f$-moments of perpetuities.

### On the rate of convergence of a regular martingale related to a branching random walk

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 326–342

Let $\mathcal{M}_n,\quad n = 1, 2, ..., $ be a supercritical branching random walk in which a number of direct descendants of an individual may be infinite with positive probability. Assume that the standard martingale $W_n$ related to $\mathcal{M}_n$ is regular, and $W$ is a limit random variable. Let $a(x)$ be a nonnegative function which regularly varies at infinity, with exponent greater than —1. We present sufficient conditions of almost sure convergence of the series $\sum^{\infty}_{n=1}a(n)(W - W_n)$. We also establish a criteria of finiteness of $EW \ln^+Wa(ln+W)$ and $EW \ln^+|Z_{\infty}|a(ln+|Z_{\infty}|)$, where $Z_{\infty} = Q_1 + \sum^{\infty}_{n=2}M_1 ... M_nQ_{n+1}$ and $(M_n, Q_n)$ are independent identically distributed random vectors, not necessarily related to $\mathcal{M}_n$.