Athreya, K. B. ; Dai, Jack
(2000)
*Random logistic maps I*
Journal of Theoretical Probability, 13
(2).
pp. 595-608.
ISSN 0894-9840

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Official URL: http://www.springerlink.com/content/ht118885410675...

Related URL: http://dx.doi.org/10.1023/A:1007828804691

## Abstract

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+1} X _{n}(1-X_{n}). It is shown here that: (i) E ln C_{1}<0⇒X_{ n}→0 w.p.1. (ii) E ln C _{1}=0⇒X _{n}→0 in probability (iii) E ln C _{1}>0, E |ln(4-C_{ 1})| < ∞ ⇒ There exists a probability measure π such that π(0, 1)=1 and π is invariant for {X_{ n}}. (iv) If there exits an invariant probability measure π such that π{0}=0, then E ln C _{1}>0 and -∫ ln(1-x) π (dx)=E ln C _{1}. (v) E ln C _{1}>0, E |ln(4-C _{1})| < ∞ 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 X_{0}≠ 0.

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

Keywords: | Random Logistic Maps; Invariant Measure |

ID Code: | 1146 |

Deposited On: | 05 Oct 2010 12:52 |

Last Modified: | 12 May 2011 09:50 |

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