The Annals of Applied Probability

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

K. B. Athreya

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Abstract

Let $\{Z_n\}^\infty_0$ be a Galton-Watson branching process with offspring distribution $\{p_j\}^\infty_0$. We assume throughout that $p_0 = 0, p_j \neq 1$ for any $j \geq 1$ and $1 < m = \Sigma jp_j < \infty$. Let $W_n = Z_nm^{-m}$ and $W = \lim_nW_n$. In this paper we study the rates of convergence to zero as $n \rightarrow \infty$ 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 $\{p_j\}$. It is shown that the rate for the first one is geometric if $p_1 > 0$ and supergeometric if $p_1 = 0$, while the rates for the other two are always supergeometric under a finite moment generating function hypothesis.

Article information

Source
Ann. Appl. Probab., Volume 4, Number 3 (1994), 779-790.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1177004971

Digital Object Identifier
doi:10.1214/aoap/1177004971

Mathematical Reviews number (MathSciNet)
MR1284985

Zentralblatt MATH identifier
0806.60068

JSTOR
links.jstor.org

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60F10: Large deviations

Keywords
Large deviation branching processes

Citation

Athreya, K. B. Large Deviation Rates for Branching Processes--I. Single Type Case. Ann. Appl. Probab. 4 (1994), no. 3, 779--790. doi:10.1214/aoap/1177004971. https://projecteuclid.org/euclid.aoap/1177004971


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See also

  • Part II: K. B. Athreya, A. N. Vidyashankar. Large Deviation Rates for Branching Processes. II. The Multitype Case. Ann. Appl. Probab., Volume 5, Number 2 (1995), 566--576.