Random Logistic Maps. I

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+1X_n(1–X_n). It is shown here that: (i) E ln C₁<0X_n0 w.p.1. (ii) E ln C₁=0X_n0 in probability (iii) E ln C₁>0, E |ln(4–C₁)|<There exists a probability measure such that (0, 1)=1 and is invariant for {X_n}. (iv) If there exits an invariant probability measure such that {0}=0, then E ln C₁>0 and – ln(1–x) (dx)=E ln C₁. (v) E ln C₁>0, E |ln(4–C₁)|< 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 X₀0.