Convergence and simulation of centred kernel quadratic stochastic operators

Abstract

In this work, we consider a class of centred kernel quadratic stochastic operators. We prove that in this class a centred kernel quadratic stochastic operator convergences almost surely and in $L^{2}$ with an exponential $L^{2}$-rate to its limit distribution. We propose an approximation scheme for this class of quadratic stochastic operators and describe three algorithms for simulating them. We consider in detail an example where the kernel is a Guassian one.