Stochastic perturbations of iterations of a simple, non-expanding,\n nonperiodic, piecewise linear, interval-map

2016/06/02 by Thomas Kaijser, Kaijser, Thomas
Computer Science · Economics, Econometrics and Finance · Mathematics · #37E05 #37H10 #60B10 #60J05 #Data Management and Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1606.00741

openalex publication_date 2016/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let g(x)=x/2 + 17/30 (mod 1), let \ξ_i, i= 1,2,... be a sequence of\nindependent, identically distributed random variables with uniform distribution\non the interval [0,1/15], define g_i(x)=g(x)+ \ξ_i (mod 1) and, for n=1,2,...,\ndefine gⁿ(x)=g_n(g_n-1(...(g₁(x))...)). For x \∈ [0,1) let \μ_n,x\ndenote the distribution of gⁿ(x). The purpose of this note is to show that\nthere exists a unique probability measure \μ, such that, for all x \∈ [0,1),\n\μ_n,x tends to \μ, as n tends to infinity. This contradicts a claim by\nLasota and Mackey from 1987 stating that the process has an asymptotic\nthree-periodicity.\n

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