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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Official URL: http://www.springerlink.com/content/w730q312017046...

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 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 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_{e}X_{e}=1-1/ϒ)=1for 1≤ϒ≤2 (b) P_{x}(lim_{e}X_{e}=1-1/ϒ)≥1 for 2 ≤ ϒ ≤4 (c)P_{x}(lim_{e}X_{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-1}_{0}I(X_{j}∈ ·) converges weakly to δ _{0}, the delta measure at 0, w.p.1 for any initial distribution of X _{0}.

Item Type: | Article |
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Source: | Copyright of this article belongs to Springer-Verlag. |

Keywords: | Random Logistic Maps; Critical Case; Convergence in Probability but not w.p.1; Empirical Distribution |

ID Code: | 1148 |

Deposited On: | 05 Oct 2010 12:51 |

Last Modified: | 12 May 2011 09:48 |

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