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

  • O. M. Iksanov


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 $\mathcal{M}_n$.
How to Cite
Iksanov, O. M. “On the Rate of Convergence of a Regular Martingale Related to a Branching Random Walk”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, no. 3, Mar. 2006, pp. 326–342, https://umj.imath.kiev.ua/index.php/umj/article/view/3457.
Research articles