Large Deviation Rates for Branching Processes--I. Single Type Case

Let {Zn}∞ 0be a Galton-Watson branching process with offspring distribution {pj}∞ 0. We assume throughout that p0= 0, pj≠ 1 for any j ≥ 1 and $1 < m = \Sigma jp_j < \infty$. Let Wn= Znm -mand$W = \lim_nW_n$. In this paper we study the rates of convergence to zero as n → ∞ of $P\big(\big|\frac{...

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Bibliographic Details
Published in:The Annals of applied probability 1994-08, Vol.4 (3), p.779-790
Main Author: Athreya, K. B.
Format: Article
Language:English
Subjects:
60F10
60J80
branching processes
Generating function
Large deviation
Perceptron convergence procedure
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Summary:Let {Zn}∞ 0be a Galton-Watson branching process with offspring distribution {pj}∞ 0. We assume throughout that p0= 0, pj≠ 1 for any j ≥ 1 and $1 < m = \Sigma jp_j < \infty$. Let Wn= Znm -mand$W = \lim_nW_n$. In this paper we study the rates of convergence to zero as n → ∞ of $P\big(\big|\frac{Z_{n+1}}{Z_n} - m\big| > \varepsilon\big),\quad P(|W_n - W| > \varepsilon)$, $P\big(\big|\frac{Z_{n+1}}{Z_n} - m \mid > \varepsilon\big|W \geq a\big)$ for $\varepsilon > 0$ and $a > 0$ under various moment conditions on {pj}. It is shown that the rate for the first one is geometric if $p_1 > 0$ and supergeometric if p1= 0, while the rates for the other two are always supergeometric under a finite moment generating function hypothesis.
ISSN:1050-5164
2168-8737
DOI:10.1214/aoap/1177004971