# 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 language English (US) 221-230 10 Journal of Multivariate Analysis 14 2 https://doi.org/10.1016/0047-259X(84)90007-1 Published - Jan 1 1984

### Fingerprint

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.
@article{3a9c7c05f7c740ac80294eabb535773d,
title = "Limiting distributions of two random sequences",
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).",
keywords = "Beta distribution, characteristic function, confluent hypergeometric function, convexity, differentiability, moment generating function, uniform random variable",
author = "Robert Chen and Richard Goodman and Alan Zame",
year = "1984",
month = "1",
day = "1",
doi = "10.1016/0047-259X(84)90007-1",
language = "English (US)",
volume = "14",
pages = "221--230",
journal = "Journal of Multivariate Analysis",
issn = "0047-259X",
number = "2",

}

TY - JOUR

T1 - Limiting distributions of two random sequences

AU - Chen, Robert

AU - Goodman, Richard

AU - Zame, Alan

PY - 1984/1/1

Y1 - 1984/1/1

N2 - 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).

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).

KW - Beta distribution

KW - characteristic function

KW - confluent hypergeometric function

KW - convexity

KW - differentiability

KW - moment generating function

KW - uniform random variable

UR - http://www.scopus.com/inward/record.url?scp=0000145754&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000145754&partnerID=8YFLogxK

U2 - 10.1016/0047-259X(84)90007-1

DO - 10.1016/0047-259X(84)90007-1

M3 - Article

AN - SCOPUS:0000145754

VL - 14

SP - 221

EP - 230

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 2

ER -