## The Annals of Applied Probability

### Harmonic moments and large deviation rates for supercritical branching processes

#### Abstract

Let $\{Z_{n}, n \ge 1 \}$ be a single type supercritical Galton--Watson process with mean $EZ_{1} \equiv m$, initiated by a single ancestor. This paper studies the large deviation behavior of the sequence $\{R_n \equiv \frac{Z_{n+1}}{Z_n}\dvtx n \ge 1 \}$ and establishes a "phase transition" in rates depending on whether $r$, the maximal number of moments possessed by the offspring distribution, is less than, equal to or greater than the Schröder constant $\alpha$. This is done via a careful analysis of the harmonic moments of $Z_n$.

#### Article information

Source
Ann. Appl. Probab., Volume 13, Number 2 (2003), 475-489.

Dates
First available in Project Euclid: 18 April 2003

https://projecteuclid.org/euclid.aoap/1050689589

Digital Object Identifier
doi:10.1214/aoap/1050689589

Mathematical Reviews number (MathSciNet)
MR1970272

Zentralblatt MATH identifier
1032.60081

#### Citation

Ney, P. E.; Vidyashankar, A. N. Harmonic moments and large deviation rates for supercritical branching processes. Ann. Appl. Probab. 13 (2003), no. 2, 475--489. doi:10.1214/aoap/1050689589. https://projecteuclid.org/euclid.aoap/1050689589

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• MADISON, WISCONSIN 53706 DEPARTMENT OF STATISTICS UNIVERSITY OF GEORGIA
• ATHENS, GEORGIA 30602 E-MAIL: anand@stat.uga.edu