# Integral representation of probabilities in Kingman coalescent

## TL;DR

This paper develops a complex integral representation for the finite time distribution of the Kingman coalescent and uses steepest descent to analyze it, leading to a local central limit theorem at small times.

## Contribution

It introduces a novel integral representation for the Kingman coalescent's finite time distribution and applies steepest descent analysis to derive a local CLT.

## Key findings

- Derived a complex integral representation for the distribution
- Applied steepest descent to analyze the integral
- Established a local central limit theorem at small times

## Abstract

Kingman Coalescent was first proposed by Kingman [7] in population genetics to describe population's genealogical structure. Now it becomes a bench-mark model for coalescent process. Extensive studies have been conducted on Kingman coalescent. In particular, its explicit finite time distribution was obtained by Tavar\'e [12]. However, very few people use this explicit distribution to do analysis for it is an intractable infinite series. In this article, we are going to establish a complex integral representation for the finite time distribution, then we use steepest descent method to analyze this integral representation to obtain local central limit theorem at small time regime.