## The Annals of Applied Probability

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

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

#### 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

#### 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.Project Euclid: euclid.aoap/1177004778