Random logistic maps II. The critical case

Abstract

Let (X _n )^∞ ₀ be a Markov chain with state space S=[0,1] generated by the iteration of i.i.d. random logistic maps, i.e., X _n+1=C _n+1 X _n (1-X _n ),n≥0, where (C _n ) ^∞₁ are i.i.d. random variables with values in [0, 4] and independent of X₀. In the critical case, i.e., when E(log C ₁)=0, Athreya and Dai(2) have shown that X _n → ^P 0. In this paper it is shown that if P(C ₁=1)<1 and E(log C ₁)=0 then (i) X _n does not go to zero with probability one (w.p.1) and in fact, there exists a 0< β < 1 and countable set Δ ⊂(0,1) such that for all x∈A:=(0,1)\Δ, P _x (X _n ≥ β for infinitely many n≥1)=1, where P _x stands for the probability distribution of (X _n ) ^∞₀ with X ₀=x w.p.1. A is a closed set for (X _n )^∞ ₀. (ii) If ϒ is the supremum of the support of the distribution of C ₁, then for all x∈A (a) P_x(lim_eX_e=1-1/ϒ)=1for 1≤ϒ≤2 (b) P_x(lim_eX_e=1-1/ϒ)≥1 for 2 ≤ ϒ ≤4 (c)P_x(lim_eX_e=ϒ/4)=1 for 2 ≤ ϒ ≤ 4 under some additional smoothness condition on the distribution of C ₁. (iii) The empirical distribution converges weakly to V _n(·)≡1/n ∑^n-1₀I(X_j∈ ·) converges weakly to δ ₀, the delta measure at 0, w.p.1 for any initial distribution of X ₀.