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What Does a Random Contingency Table Look Like?

Published online by Cambridge University Press:  12 February 2010

ALEXANDER BARVINOK
ALEXANDER BARVINOK*
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA (e-mail: [email protected])
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Abstract

Let R = (r1, . . ., rm) and C = (c1, . . ., cn) be positive integer vectors such that r1 + ⋯ + rm = c1 + ⋯ + cn. We consider the set Σ(R, C) of non-negative m × n integer matrices (contingency tables) with row sums R and column sums C as a finite probability space with the uniform measure. We prove that a random table D ∈ Σ(R, C) is close with high probability to a particular matrix (‘typical table’) Z defined as follows. We let g(x) = (x + 1)ln(x + 1) − x ln x for x ≥ 0 and let g(X) = ∑i,jg(xij) for a non-negative matrix X = (xij). Then g(X) is strictly concave and attains its maximum on the polytope of non-negative m × n matrices X with row sums R and column sums C at a unique point, which we call the typical table Z.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 4 , July 2010 , pp. 517 - 539
Copyright
Copyright © Cambridge University Press 2010

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