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A weighted random walk model, with application to a genetic algorithm

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  01 July 2016

C. Dombry
C. Dombry*
Affiliation:
Université Claude Bernard Lyon 1
*
∗ Postal address: Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France. Email address: dombry@math.univ-lyon1.fr
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Abstract

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We consider a weighted random walk model defined as follows. An n-step random walk on the integers with distribution Pn is weighted by giving the path S=(S0,…,Sn) a probability proportional to where the function f is the so-called fitness function. In the case of power-type fitness, we prove the convergence of the renormalized path to a deterministic function with exponential speed. This function is a solution to a variational problem. In the case of the simple symmetric random walk, explicit computations are done. Our result relies on large deviations techniques and Varadhan's integral lemma. We then study an application of this model to mutation-selection dynamics on the integers where a random walk operates the mutation. This dynamics is the infinite-population limit of that of mutation-selection genetic algorithms. We prove that the population grows to ∞ and make explicit its growth speed. This is a toy model for modelling the effect of stronger selection at ∞ for genetic algorithms taking place in a noncompact space.

Keywords

Random walkfunctional large deviations principlegenetic algorithm

MSC classification

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 2 , June 2007 , pp. 550 - 568
Copyright
Copyright © Applied Probability Trust 2007 

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