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RESEARCH PRODUCT

Random Logistic Maps II. The Critical Case

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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 X 0. 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 a countable set ▵⊂(0,1) such that for all x∈A≔(0,1)∖▵, 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 γ is the supremum of the support of the distribution of C 1, then for all x∈A (a) $$P_x (\overline {\lim _n } X_n = 1 - \frac{1}{\gamma }) = 1{\text{ for 1}} \leqslant \gamma \leqslant {\text{2}}$$ for 1≤γ≤2 (b) $$P_x (\overline {\lim _n } X_n \geqslant 1 - \frac{1}{\gamma }) = 1{\text{ for 2}} \leqslant \gamma \leqslant {\text{4}}$$ for 2≤γ≤4 (c) $$P_x (\overline {\lim _n } X_n = \frac{\gamma }{4}) = 1{\text{ for 2}} \leqslant \gamma \leqslant {\text{4}}$$ for 2≤γ≤4 under some additional smoothness condition on the distribution of C 1. (iii) The empirical distribution $$V_n ( \cdot ) \equiv \frac{1}{n}\sum\nolimits_0^{n - 1} {} I(X_j \in \cdot )$$ converges weakly to δ 0, the delta measure at 0, w.p.1 for any initial distribution of X 0.