On the rate of convergence of a regular martingale related to a branching random walk
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....
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| Datum: | 2006 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2006
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3457 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | 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 $\mathcal{M}_n$. |
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