Radix expansions and the uniform distribution

Rameshwar D. Gupta, Donald St P. Richards

Research output: Contribution to journalArticle

Abstract

Let the random variable X be uniformly distributed on [0,1], α be a positive number, α≠1, and b be a positive integer, b>We derive the joint distribution of Y1,Y2,...,Yk, the first k significant digits in the radix expansion in base b of Y=X1/α. We show that, as k→∞,Yk converges in distribution to the uniform distribution on the set {0,1,...,b-1}. We also prove that if Y is a random variable taking values in [0,1] whose cumulative distribution function is continuous and convex (respectively, concave) then the significant digits Y1,Y2,... are stochastically increasing (respectively, decreasing). In particular, if Y=X1/α where X is uniformly distributed on [0,1] then the significant digits Y1,Y2,... are stochastically increasing (respectively, decreasing) if α<1 (respectively, α>1).

Original languageEnglish (US)
Pages (from-to)263-270
Number of pages8
JournalStatistics and Probability Letters
Volume46
Issue number3
DOIs
Publication statusPublished - Feb 1 2000

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All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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