# Random logistic maps II. The critical case

Athreya, K. B. ; Schuh, H. -J. (2003) Random logistic maps II. The critical case Journal of Theoretical Probability, 16 (4). pp. 813-830. ISSN 0894-9840

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Related URL: http://dx.doi.org/10.1023/B:JOTP.0000011994.90898.81

## 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 nP 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) Px(limeXe=1-1/ϒ)=1for 1≤ϒ≤2 (b) Px(limeXe=1-1/ϒ)≥1 for 2 ≤ ϒ ≤4 (c)Px(limeXe=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(Xj∈ ·) converges weakly to δ 0, the delta measure at 0, w.p.1 for any initial distribution of X 0.

Item Type: Article Copyright of this article belongs to Springer-Verlag. Random Logistic Maps; Critical Case; Convergence in Probability but not w.p.1; Empirical Distribution 1148 05 Oct 2010 12:51 12 May 2011 09:48

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