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Transactions of the American Mathematical Society

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The local limit theorem and some related aspects of super-critical branching processes


Authors: Krishna B. Athreya and Peter Ney
Journal: Trans. Amer. Math. Soc. 152 (1970), 233-251
MSC: Primary 60.67
DOI: https://doi.org/10.1090/S0002-9947-1970-0268971-X
MathSciNet review: 0268971
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Abstract: Let $ \{ {Z_n}:n = 0,1,2, \ldots \} $ be a Galton-Watson branching process with offspring p.g.f. $ f(s) = \Sigma _0^\infty {p_j}{s^j}$. Assume (i) $ 1 < m = f'(1 - ) = \Sigma _1^\infty j{p_j} < \infty $, (ii) $ \Sigma _1^\infty {j^2}{p_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_n}{m^{ - n}}$ with $ {Z_0} = 1,{w^{(i)}}(x)$ the $ i$-fold convolution of $ w(x),{P_n}(i,j) = P({Z_n} = j\vert{Z_0} = i),{\delta _0} = (\log \gamma _0^{ - 1}){(\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

$\displaystyle \vert{m^n}{P_n}(i,j) - {w^{(i)}}({m^{ - n}}j)\vert \leqq C[\beta _0^{ - n}{({m^{ - n}}j)^{ - 1}} + {\beta ^{ - n}}]$

for all $ j \geqq 1$. Applications to the boundary theory of the associated space time process are also discussed.

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DOI: https://doi.org/10.1090/S0002-9947-1970-0268971-X
Keywords: Galton-Watson process, branching process, local limit theorems, potential theory, space-time process
Article copyright: © Copyright 1970 American Mathematical Society