## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 8, Number 1 (1998), 182-192.

### Elliptic and other functions in the large deviations behavior of the Wright-Fisher process

#### Abstract

The present paper continues the work of two previous papers on the variational behavior, over a large number of generations, of a Wright-Fisher process modelling an even larger reproducing population. It was shown that a Wright-Fisher process subject to random drift and one-way mutation which undergoes a large deviation follows with near certainty a path which can be a trigonometric, exponential, hyperbolic or parabolic function. Here it is shown that a process subject to random drift and gamete selection follows in similar circumstances a path which is, apart from critical cases, a Jacobian elliptic function.

#### Article information

**Source**

Ann. Appl. Probab., Volume 8, Number 1 (1998), 182-192.

**Dates**

First available in Project Euclid: 29 July 2002

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1027961039

**Digital Object Identifier**

doi:10.1214/aoap/1027961039

**Mathematical Reviews number (MathSciNet)**

MR1620354

**Zentralblatt MATH identifier**

0942.60020

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) [See also 90B30, 91D10, 91D35, 91E40]

**Keywords**

Wright-Fisher process large
deviations action functional calculus of
variations elliptic functions

#### Citation

Papangelou, F. Elliptic and other functions in the large deviations behavior of the Wright-Fisher process. Ann. Appl. Probab. 8 (1998), no. 1, 182--192. doi:10.1214/aoap/1027961039. https://projecteuclid.org/euclid.aoap/1027961039