The Annals of Applied Probability

Harmonic moments and large deviation rates for supercritical branching processes

P. E. Ney and A. N. Vidyashankar

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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 $Z_n$.

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

First available in Project Euclid: 18 April 2003

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Zentralblatt MATH identifier

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

Branching processes harmonic moments large deviations


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.

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