## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 2 (1988), 502-507.

### On the Maximum Sequence in a Critical Branching Process

#### Abstract

If $\{Z_n\}^\infty_0$ is a critical branching process such that $E_1Z^2_1 < \infty$, then $(\log n)^{-1}E_iM_n \rightarrow i$, where $E_i$ refers to starting with $Z_0 = i$ and $M_n = \max_{0\leq j \leq n}Z_j$. This improves the earlier results of Weiner [9] and Pakes [7].

#### Article information

**Source**

Ann. Probab., Volume 16, Number 2 (1988), 502-507.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991770

**Digital Object Identifier**

doi:10.1214/aop/1176991770

**Mathematical Reviews number (MathSciNet)**

MR929060

**Zentralblatt MATH identifier**

0643.60063

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Secondary: 60K99: None of the above, but in this section

**Keywords**

Branching process critical maximum

#### Citation

Athreya, K. B. On the Maximum Sequence in a Critical Branching Process. Ann. Probab. 16 (1988), no. 2, 502--507. doi:10.1214/aop/1176991770. https://projecteuclid.org/euclid.aop/1176991770