Centred quadratic stochastic operators

TL;DR

This paper investigates the weak convergence of iterates of centred kernel quadratic stochastic operators, providing conditions for convergence, an efficient simulation method, and insights into their behavior in population models.

Contribution

It establishes weak convergence results under mild assumptions and introduces a simulation algorithm for quadratic stochastic operators based on sums of random variables.

Findings

Weak convergence achieved under finite variance or tail control.

An efficient simulation algorithm for the iterates.

Insights into intrinsic difficulties in analyzing quadratic stochastic operators.

Abstract

We study the weak convergence of iterates of so-called centred kernel quadratic stochastic operators. These iterations, in a population evolution setting, describe the additive perturbation of the arithmetic mean of the traits of an individual's parents and correspond to certain weighted sums of independent random variables. We show that one can obtain weak convergence results under rather mild assumptions on the kernel. Essentially it is sufficient for the distribution of the perturbing random variable to have a finite variance or have tails controlled by a power function. The advantage of these conditions is that in many cases they are easily verifiable by an applied user. Additionally, the representation by sums of random variables implies an efficient simulation algorithm to obtain random variables approximately following the law of the iterates of the quadratic stochastic operator, with full control of the degree of approximation. Our results also indicate where lies an intrinsic difficulty in the analysis of the behaviour of quadratic stochastic operators.