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Abstract
In this paper, the convergence rates of harmonic moment E[(1 · Zn)−r] of a supercritical multi-type Galton–Watson process {Zn; n ≥ 0} are studied. It is shown that there exists a phase transition in the convergence rate of the harmonic moments, which extends the results from single type cases to multi-type cases. Based on above result, under some moment hypothesis on Zn, the large deviations of $\boldsymbol{l} \cdot \boldsymbol{Z}_{n+1}\over \boldsymbol{1}\cdot\boldsymbol{Z}_{n}$ are studied for l > 0. In particular, in the case that the offspring distributions are of Pareto-type, an explicit large deviation result is obtained.
Keywords
Supercritical
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multi-type
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Galton–Watson processes
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harmonic moments
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large deviations
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Jiangrui Tan, Mei Zhang.
Harmonic Moments for Supercritical Multi-type Galton–Watson Processes.
Frontiers of Mathematics 1-26 DOI:10.1007/s11464-022-0129-8
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