Limiting distributions of two random sequences

Robert Chen, Richard Goodman, Alan Zame

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

For fixed p (0 ≤ p ≤ 1), let {L0, R0} = {0, 1} and X1 be a uniform random variable over {L0, R0}. With probability p let {L1, R1} = {L0, X1} or = {X1, R0} according as X1 ≥ 1 2(L0 + R0) or < 1 2(L0 + R0); with probability 1 - p let {L1, R1} = {X1, R0} or = {L0, X1} according as X1 ≥ 1 2(L0 + R0) or < 1 2(L0 + R0), and let X2 be a uniform random variable over {L1, R1}. For n ≥ 2, with probability p let {Ln, Rn} = {Ln - 1, Xn} or = {Xn, Rn - 1} according as Xn ≥ 1 2(Ln - 1 + Rn - 1) or < 1 2(Ln - 1 + Rn - 1), with probability 1 - p let {Ln, Rn} = {Xn, Rn - 1} or = {Ln - 1, Xn} according as Xn ≥ 1 2(Ln - 1 + Rn - 1) or < 1 2(Ln - 1 + Rn - 1), and let Xn + 1 be a uniform random variable over {Ln, Rn}. By this iterated procedure, a random sequence {Xn}n ≥ 1 is constructed, and it is easy to see that Xn converges to a random variable Yp (say) almost surely as n → ∞. Then what is the distribution of Yp? It is shown that the Beta, (2, 2) distribution is the distribution of Y1; that is, the probability density function of Y1 is g(y) = 6y(1 - y) I0,1(y). It is also shown that the distribution of Y0 is not a known distribution but has some interesting properties (convexity and differentiability).

Original languageEnglish (US)
Pages (from-to)221-230
Number of pages10
JournalJournal of Multivariate Analysis
Volume14
Issue number2
DOIs
StatePublished - Jan 1 1984

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Random Sequence
Limiting Distribution
Random variables
Random variable
Probability density function
Differentiability
Convexity
Limiting distribution
Converge

Keywords

  • Beta distribution
  • characteristic function
  • confluent hypergeometric function
  • convexity
  • differentiability
  • moment generating function
  • uniform random variable

ASJC Scopus subject areas

  • Statistics, Probability and Uncertainty
  • Numerical Analysis
  • Statistics and Probability

Cite this

Limiting distributions of two random sequences. / Chen, Robert; Goodman, Richard; Zame, Alan.

In: Journal of Multivariate Analysis, Vol. 14, No. 2, 01.01.1984, p. 221-230.

Research output: Contribution to journalArticle

Chen, Robert ; Goodman, Richard ; Zame, Alan. / Limiting distributions of two random sequences. In: Journal of Multivariate Analysis. 1984 ; Vol. 14, No. 2. pp. 221-230.
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AB - For fixed p (0 ≤ p ≤ 1), let {L0, R0} = {0, 1} and X1 be a uniform random variable over {L0, R0}. With probability p let {L1, R1} = {L0, X1} or = {X1, R0} according as X1 ≥ 1 2(L0 + R0) or < 1 2(L0 + R0); with probability 1 - p let {L1, R1} = {X1, R0} or = {L0, X1} according as X1 ≥ 1 2(L0 + R0) or < 1 2(L0 + R0), and let X2 be a uniform random variable over {L1, R1}. For n ≥ 2, with probability p let {Ln, Rn} = {Ln - 1, Xn} or = {Xn, Rn - 1} according as Xn ≥ 1 2(Ln - 1 + Rn - 1) or < 1 2(Ln - 1 + Rn - 1), with probability 1 - p let {Ln, Rn} = {Xn, Rn - 1} or = {Ln - 1, Xn} according as Xn ≥ 1 2(Ln - 1 + Rn - 1) or < 1 2(Ln - 1 + Rn - 1), and let Xn + 1 be a uniform random variable over {Ln, Rn}. By this iterated procedure, a random sequence {Xn}n ≥ 1 is constructed, and it is easy to see that Xn converges to a random variable Yp (say) almost surely as n → ∞. Then what is the distribution of Yp? It is shown that the Beta, (2, 2) distribution is the distribution of Y1; that is, the probability density function of Y1 is g(y) = 6y(1 - y) I0,1(y). It is also shown that the distribution of Y0 is not a known distribution but has some interesting properties (convexity and differentiability).

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KW - confluent hypergeometric function

KW - convexity

KW - differentiability

KW - moment generating function

KW - uniform random variable

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