Random Logistic Maps. I

Let {C i}(infinity) 0 be a sequence of independent and identically distributed random variables with vales in [0, 4]. Let {X n}(infinity) 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 (longrightarrow) X n(longrightarrow) 0 w.p.1. (ii) E ln C 1=0 (longrightarrow) X n(longrightarrow) 0 in probability (iii) E ln C 1>0, E |ln(4-C 1)|< (infinity) (longrightarrow) There exists a probability measure (pi) such that (pi)(0, 1)=1 and (pi) is invariant for {X n}. (iv) If there exits an invariant probability measure (pi) such that (pi) {0}=0, then E ln C 1>0 and – (integral) ln(1–x) (pi)(dx)=E ln C 1. (v) E ln C 1>0, E |ln(4–C 1)|< (infinity) 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 (not equal) 0.