### Abstract

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

Original language | English (US) |
---|---|

Pages (from-to) | 221-230 |

Number of pages | 10 |

Journal | Journal of Multivariate Analysis |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 1984 |

### Fingerprint

### 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

*Journal of Multivariate Analysis*,

*14*(2), 221-230. https://doi.org/10.1016/0047-259X(84)90007-1

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

Research output: Contribution to journal › Article

*Journal of Multivariate Analysis*, vol. 14, no. 2, pp. 221-230. https://doi.org/10.1016/0047-259X(84)90007-1

}

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 -