The Annals of Probability

Pairwise Independent Random Variables

G. L. O'Brien

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Abstract

Let $Y_1, \cdots, Y_r$ be independent random variables, each uniformly distributed on $\mathscr{M} = \{1,2, \cdots, M\}$. It is shown that at most $N = 1 + M + \cdots + M^{r-1}$ pairwise independent random variables, all uniform on $\mathscr{M}$ and all functions of $(Y_1, \cdots, Y_r)$, can be defined. If $M = p^k$ for some prime $p$, the maximum can be attained by a strictly stationary sequence $X_1, \cdots, X_N$, for which any $r$ successive random variables are independent.

Article information

Source
Ann. Probab., Volume 8, Number 1 (1980), 170-175.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994834

Digital Object Identifier
doi:10.1214/aop/1176994834

Mathematical Reviews number (MathSciNet)
MR556424

Zentralblatt MATH identifier
0426.60011

JSTOR
links.jstor.org

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 65C10: Random number generation 60B99: None of the above, but in this section 62K10: Block designs

Keywords
Pairwise independence stationary sequences pseudorandom numbers block designs

Citation

O'Brien, G. L. Pairwise Independent Random Variables. Ann. Probab. 8 (1980), no. 1, 170--175. doi:10.1214/aop/1176994834. https://projecteuclid.org/euclid.aop/1176994834


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