Random logistic maps I

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


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-Xn). It is shown here that: (i) E ln C1<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 X0≠ 0.

Item Type:Article
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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