Open Access
Translator Disclaimer
January 2002 On the Nonuniqueness of the Invariant Probability for I.I.D. Random Logisitc Maps
K.B. Athreya, J.J. Dai
Ann. Probab. 30(1): 437-442 (January 2002). DOI: 10.1214/aop/1020107774

Abstract

Let $\{X_n\}^{\infty}_0$ be a Markov chain with values in $[0,1]$ generated by the iteration of random logistic maps defined by $X_{n+1}=f_{C_{n+1}}(X_n)\equiv C_{n+1}X_n(1-X_n)$, $n=0,1,2,\ldots\,$, with $\{C_n\}^{\infty}_1$ being independent and identically distributed random variables with values in $[0,4]$ and independent of $X_0$. This paper provides a class of examples where $C_i$ take only two values $\lambda$ and $\mu$ such that there exist two distinct invariant probability distributions $\pi_0$ and $\pi_1$ supported by the open interval $(0,1)$. This settles a question raised by R. N. Bhattacharya.

Citation

Download Citation

K.B. Athreya. J.J. Dai. "On the Nonuniqueness of the Invariant Probability for I.I.D. Random Logisitc Maps." Ann. Probab. 30 (1) 437 - 442, January 2002. https://doi.org/10.1214/aop/1020107774

Information

Published: January 2002
First available in Project Euclid: 29 April 2002

Digital Object Identifier: 10.1214/aop/1020107774

Subjects:
Primary: 60J05 , 92D25
Secondary: 60F05

Keywords: invariant probability , logistic maps , uniqueness

Rights: Copyright © 2002 Institute of Mathematical Statistics

JOURNAL ARTICLE
6 PAGES


SHARE
Vol.30 • No. 1 • January 2002
Back to Top