The Local Limit Theorem and Some Related Aspects of Super-Critical Branching Processes

The Local Limit Theorem and Some Related Aspects of Super-Critical Branching Processes

Let $\{Z_n: n = 0, 1, 2,\ldots\}$ be a Galton-Watson branching process with offspring p.g.f. $f(s) = \sum^\infty_0 p_jS^j$. Assume (i) $1 < m = f'(1-) = \sum^\infty_1 jp_j < \infty$, (ii) $\sum^infty_1 j^2p_j < \infty$ and (iii) $\gamma_0 = f'(q) > 0$, where $q$ is the extincti...

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Veröffentlicht in:Transactions of the American Mathematical Society. - American Mathematical Society, 1900. - 152(1970), 1, Seite 233-251
1. Verfasser: Athreya, Krishna B. (VerfasserIn)
Weitere Verfasser: Ney, Peter
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 1970
Zugriff auf das übergeordnete Werk:Transactions of the American Mathematical Society
Schlagworte:Galton-Watson proces branching process local limit theorems potential theory space-time process Mathematics Arts Physical sciences Law
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520 |a Let $\{Z_n: n = 0, 1, 2,\ldots\}$ be a Galton-Watson branching process with offspring p.g.f. $f(s) = \sum^\infty_0 p_jS^j$. Assume (i) $1 < m = f'(1-) = \sum^\infty_1 jp_j < \infty$, (ii) $\sum^infty_1 j^2p_j < \infty$ and (iii) $\gamma_0 = f'(q) > 0$, where $q$ is the extinction probability of the process. Let $w(x)$ denote the density function of $W$, the almost sure limit of $Z_nm^{-n}$ with $Z_0 = 1, w^{(i)}(x)$ the $i$-fold convolution of $w(x), P_n(i,j) = P(Z_n = j \mid Z_0 = i), \delta_0 = (\log \gamma^{-1}_0)(\log m)^{-1}$ and $\beta_0 = m^{\delta_0/(3 + \delta_0)}$. Then for any $0 < \beta < beta_0$ and $i$ we can find a constant $C = C(i,\beta)$ such that $$|m^nP_n(i, j)-w^{(i)}(m^{-n}j)| \leqq C \lbrack\beta^{-n}_0(m^{-n}j)^{-1} + \beta^{-n}\rbrack$$ for all $j \geqq 1$. Applications to the boundary theory of the associated space time process are also discussed. 
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