Limiting distributions of two random sequences
type: Item
instance of: scholarly article
title: Limiting distributions of two random sequences
zbMATH Open document ID: 0537.62014
DOI: 10.1016/0047-259X(84)90007-1
author: N. E. Zubov
published in: Journal of Multivariate Analysis
publication date: 1984-01-01T00:00:00Z
review text: Let p (0\(\leq p\leq 1)\) be fixed. We define a sequence of random intervals \(\{[L_ n,R_ n]\), \(n\geq 0\}\) and a sequence of random variables \(\{X_ n\), \(n\geq 1\}\), inductively, as follows: Firstly, let \([L_ 0,R_ 0]=[0,1]\). Secondly, let \([L_{n-1},R_{n-1}]\) be defined. Let \(X_ n\) be a uniform random variable over \([L_{n-1},R_{n-1}]\) and put with probability p \[ [L_ n,R_ n]= \begin{cases} [L_{n-1},X_ n]&\;if\;X_ n\geq frac{1}{2}(L_{n-1}+R_{n-1}), \\ [X_ n,R_{n-1}]&\;if\;X_ n<frac{1}{2}(L_{n-1}+R_{n-1}), \end{cases} \] and with probability \(1-p\) \[ [L_ n,R_ n]= \begin{cases} [L_{n-1},X_ n]&\;if\;X_ n<frac{1}{2}(L_{n-1}+R_{n-1}), \\ [X_ n,R_{n-1}]&\;if\;X_ n\geq frac{1}{2}(L_{n-1}+R_{n- 1}).\end{cases} \] It is easy to see that \(X_ n\) converges to a random variable \(Y_ p\) (say) almost surely as \(n\to \infty\). In this paper, the author considered the distribution of \(Y_ 1\) or \(Y_ 0\) and showed that the distribution of \(Y_ 1\) is the beta (2,2) distribution.
Mathematics Subject Classification ID: 62E20
Mathematics Subject Classification ID: 62E10
zbMATH DE Number: 3854188
zbMATH Keywords: beta distribution
zbMATH Keywords: convexity
zbMATH Keywords: differentiability
zbMATH Keywords: moment generating function
zbMATH Keywords: confluent hypergeometric function
zbMATH Keywords: sequence of random intervals
zbMATH Keywords: uniform random variable
reviewed by: Ken-ichi Yoshihara
MaRDI profile type: Publication
cites work: Q5545069
cites work: The asymptotic behavior of spacings under Kakutani's model for interval subdivision
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cites work: Q4160140
cites work: A proof of Kakutani's conjecture on random subdivision of longest intervals
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full work available at URL: 0047-259x(84)90007-1
OpenAlex ID: W2012665816
description: scientific article; zbMATH DE number 3854188