Moments and Distributions of Trajectories in Slow Random Monads

摘要

Given a finite set B (basin) with n1 elements, which we call points, and a map M:B→B, we call such pairs (B,M) monads. Here we study a class of random monads, where the values of M(⋅) are independently distributed in B as follows: for all a,b∈B the probability of M(a)=a is s and the probability of M(a)=b, where a≠b, is (1−s)/(n−1). Here s is a parameter, 0≤s≤1. We fix a point ⊙∈B and consider the sequence M t (⊙), t=0,1,2,… . A point is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis n , Rec n and Tra n the numbers of visited, recurrent and transient points respectively. We prove that, when n tends to infinity, Vis n and Tra n converge in law to geometric distributions and Rec n converges in law to a distribution concentrated at its lowest value, which is one. Now about moments. The case s=1 is trivial, so let 0≤s1. For any natural number k there is a number such that the k-th moments of Vis n , Rec n and Tra n do not exceed this number for all n. About Vis n : for any natural k the k-th moment of Vis n is an increasing function of n. So it has a limit when n→∞ and for all n it is less than this limit. About Rec n : for any k the k-th moment of Rec n tends to one when n tends to infinity. About Tra n : for any k the k-th moment of Tra n has a limit when n tends to infinity.