Abstract
In this paper, we study the small-time asymptotic behavior of the Kingman coalescent. We obtain the Berry-Esseen bound and the Edgeworth expansion in the central limit theorem. Moreover, by the method of mod-ϕ convergence, we also obtain the precise large deviations and the precise moderate deviations. Last, we also obtain a non-asymptotic deviation inequality for the Kingman coalescent.
Highlights
The Kingman coalescent was introduced in 1982 by Kingman in [5]
We study the small-time asymptotic behavior of the Kingman coalescent
The Kingman coalescent usually describes the backward movement in time in population genetics
Summary
The Kingman coalescent was introduced in 1982 by Kingman in [5]. It was used to describe the genealogy of a sample from a population. The Kingman coalescent usually describes the backward movement in time in population genetics. Apart from describing sample genealogy, the Kingman coalescent has other interpretations, such as the number of surviving ancient families. Once an ancient family disappears, the Kingman coalescent experiences a coalescing event These coalescing events arrive independently, and waiting times between coalescing events follow exponential distributions. Let Tn be the arriving time of the coalescing event, where n is the number of surviving ancient families right after Tn. Tn =. We have not seen any asymptotic result like the Berry-Esseen bounds, the Edgeworth expansions and deviation inequalities for the Kingman coalescent in the existing literatures either. Precise moderate deviations, and deviation inequalities are obtained for the Kingman coalescent.
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