Random Logistic Maps. I

Abstract

Let {C _i}^∞ ₀ be a sequence of independent and identically distributed random variables with vales in [0, 4]. Let {X _n}^∞ ₀ be a sequence of random variables with values in [0, 1] defined recursively by X _n+1=C _n+1 X _n(1−X _n). It is shown here that: (i) E ln C ₁<0⇒X _n→0 w.p.1. (ii) E ln C ₁=0⇒X _n→0 in probability (iii) E ln C ₁>0, E |ln(4−C ₁)|<∞⇒There exists a probability measure π such that π(0, 1)=1 and π is invariant for {X _n}. (iv) If there exits an invariant probability measure π such that π{0}=0, then E ln C ₁>0 and −∫ ln(1−x) π(dx)=E ln C ₁. (v) E ln C ₁>0, E |ln(4−C ₁)|<∞ and {X _n} is Harris irreducible implies that the probability distribution of X _n converges in the Cesaro sense to a unique probability distribution on (0, 1) for all X ₀≠0.