Abstract
Let ( Z n ) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z n / E [ Z n | ξ ] . We show large and moderate deviation principles for the sequence log Z n (with appropriate normalization) by finding an equivalence of the moments of Z n and a criterion for the existence of harmonic moments of W.
Highlights
A branching process in a random environment (BPRE) is a natural and important generalisation of the Galton-Watson process, where the reproduction law varies according to a random environment indexed by time
For background concepts and basic results concerning a BPRE we refer to Athreya and Karlin [4, 3]
In the critical and subcritical regime the branching process goes out and the research interest has been mostly concentrated on the survival probability and conditional limit theorems, see e.g. Afanasyev, Böinghoff, Kersting, Vatutin [1, 2], Vatutin [26], Vatutin and Zheng [27], and the references therein
Summary
A branching process in a random environment (BPRE) is a natural and important generalisation of the Galton-Watson process, where the reproduction law varies according to a random environment indexed by time. Random environment, harmonic moments, large deviations, phase transitions, central limit theorem. C (r) C (r) C (r) if r > r1, if r = r1, if r < r1, where C(r) are positive constants for which we find integral expressions This shows that there are three phase transitions in the rate of convergence of the harmonic moments for the process Zn, with the critical value r1. It generalizes the result of [23] for the Galton-Watson process. In the critical case where r = r1, the limit constant obtained in this paper is different to that of [23, Theorem 1], which leads to an alternative expression of the constant and the following nice identity involving the well-known functions G, Q and φ: defining. Throughout the paper, we denote by C an absolute constant whose value may differ from line to line
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