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Pré-Publication, Document De Travail Année : 2020

Cramér type moderate deviation expansion for a supercritical multi-type branching process in a random environment

Résumé

Let $Z_n^i=(Z_n^i(1), \cdots, Z_n^i(d)),\,n\geq 0,$ be a supercritical $d$-type branching process in an independent and identically distributed random environment $\xi =(\xi_0, \xi_1, \cdots)$, starting with one initial particle of type $i$. We establish a Cram\'er type moderate deviation expansion for $\log\|Z_n^i\|$. To this end, we first prove uniform results for the existence of harmonic moments and Berry-Esseen type bound under suitably changed measure, using the spectral gap theory on products of random matrices and the fundamental martingale that we found in an earlier paper.
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Dates et versions

hal-02934081 , version 1 (08-09-2020)

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  • HAL Id : hal-02934081 , version 1

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Ion Grama, Quansheng Liu, Erwan Pin. Cramér type moderate deviation expansion for a supercritical multi-type branching process in a random environment. 2020. ⟨hal-02934081⟩
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