Moments and Distributions of Trajectories in Slow Random Monads

Given a finite set B (basin) with n >1 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≤ s <1. 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.