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The Residues modulo m of Products of Random Integers

Published online by Cambridge University Press:  09 October 2002

WOLFGANG STADJE
WOLFGANG STADJE
Affiliation:
Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany (e-mail: wolfgang@mathematik.uni-osnabrueck.de)
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Abstract

For two stochastically dependent random variables X and Y taking values in {0,…, m−1}, we study the distribution of the random residue U = XY mod m. Our main result is an upper bound for the distance Δm = supx∈[0,1] [mid ] P(U/m [les ] x)−x[mid ]. For independent and uniformly distributed X and Y, the exact distribution of U is derived and shown to be stochastically smaller than the uniform distribution on {0,…, m−1}. Moreover, in this case Δm is given explicitly.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 11 , Issue 5 , September 2002 , pp. 529 - 540
Copyright
2002 Cambridge University Press

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