Elementary Symmetric Polynomials in Random Variables
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Elementary Symmetric Polynomials in Random Variables Aivars Lorencs
Published online: 23 March 2007 © Springer Science + Business Media B.V. 2007
Abstract The subject of the paper is the probability-theoretic properties of elementary symmetric polynomials σk of arbitrary degree k in random variables Xi (i = 1, 2, . . . , m) defined on special subsets of commutative rings Rm with identity of finite characteristic m. It is shown that the probability distributions of the random elements σk (X1 , . . . , Xm ) tend to a limit when m → ∞ if X1 , . . . , Xm form a Markov chain of finite degree μ over a finite set of states V , V ⊂ Rm , with positive conditional probabilities. Moreover, if all the conditional probabilities exceed a prescribed positive number α, the limit distributions do not depend on the choice of the chain. Keywords Elementary symmetric polynomial · Commutative ring with identity · Regular permutational automaton · Complex Markov chain Mathematics Subject Classification (2000) Primary 13M10 · Secondary 60B12 · 68Q45
1 Introduction Probability-theoretic properties of elementary symmetric polynomials in random variables was the objective of [1, 2, 5]. In the paper [2], studied were the probability-theoretic properties of polynomials σ1 in random variables defined on Abelian groups of natural numbers with respect to addition modulo M, M ≥ 2. In [5], elementary symmetric polynomials σ1 in random variables were considered on arbitrary finite Abelian groups. On the contrary, Denmead Smith [1] studied properties of elementary symmetric polynomials σk of arbitrary degree k ≥ 1; however, the random variables Xi in his paper were defined on Galois fields. In addition, all the variables were supposed to be mutually independent and having equal distributions. The main tool of Denmead Smith was the multidimensional Fourier analysis, A. Lorencs () Institute of Electronics and Computer Science, University of Latvia, 14 Dzerbenes Str., 1006 Riga, Latvia e-mail: [email protected]
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which helped him to obtain a number of results concerning the structure of the considered limiting distributions and speed of convergence of the distributions to the limit. In the present work, the apparatus of permutational Moore automata (see [3, 4]) is used as the main tool of analysis of the probability-theoretic properties of the polynomials σk . Definition 1 A finite deterministic automaton defined by the input alphabet X = {x1 , x2 , . . . , xl }, output alphabet Y = {y1 , y2 , . . . , ys }, set of inner states Z = {z1 , z2 , . . . , zn }, transfer function from Z × X to Z, and output function λ from Z × X onto Y is said to be a Moore automaton if there is a surjection μ: Z → Y such that, for all pairs zx ∈ Z × X, λ(z, x) = μ(z, x). Definition 2 Let us call a Moore automaton X, Y, Z, , λ a permutation automaton if (1) for every x ∈ X, the function determines a permutation on Z: z1 z2 ··· zn , (z1 , x) (z2 , x) · · · (zn , x) and (2) (z, x) = (z, x ) for x = x . Denoting by Ax the (n × n)-permutation matrix w
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