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.
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Athreya, K.B., Dai, J. Random Logistic Maps. I. Journal of Theoretical Probability 13, 595–608 (2000). https://doi.org/10.1023/A:1007828804691
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DOI: https://doi.org/10.1023/A:1007828804691