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On uniqueness of the problem of moments in the class of -distributions

On uniqueness of the problem of moments in the class of -distributions -- We introduce the notion of q - (q\, ...,g,)-binomial invariants of a random vector = (, ...,/) distributed on the set of vectors whose /th coordinate is a non-negative integer power of the number q, > 1 , / = 1, ...,/. We find relations between ^-binomial invariants and mixed moments, give conditions under which the distribution of the vector is uniquely determined by the sequence of its ^-binomial invariants, and present expressions of probabilities (, = ?;·,..., , =tf'), r,,. ...0 = 0,1,..., in terms of the corresponding ^-binomial invariants and estimates of these probabilities. We prove a theorem on convergence of a sequence of distributions of such random vectors under the condition that the corresponding ^-binomial invariants converge. The results obtained are used in the study of the limit distribution of the number of solutions of a class of systems of linear equations over a finite field. INTRODUCTION In [2-4], the class of systems of random linear equations over a finite field or a ring R of cardinality q of the form } = °> / = , . . . , + J, (1) 7=1 is considered; here s is a fixed integer, a^ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

On uniqueness of the problem of moments in the class of -distributions

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1998.8.1.1
Publisher site
See Article on Publisher Site

Abstract

-- We introduce the notion of q - (q\, ...,g,)-binomial invariants of a random vector = (, ...,/) distributed on the set of vectors whose /th coordinate is a non-negative integer power of the number q, > 1 , / = 1, ...,/. We find relations between ^-binomial invariants and mixed moments, give conditions under which the distribution of the vector is uniquely determined by the sequence of its ^-binomial invariants, and present expressions of probabilities (, = ?;·,..., , =tf'), r,,. ...0 = 0,1,..., in terms of the corresponding ^-binomial invariants and estimates of these probabilities. We prove a theorem on convergence of a sequence of distributions of such random vectors under the condition that the corresponding ^-binomial invariants converge. The results obtained are used in the study of the limit distribution of the number of solutions of a class of systems of linear equations over a finite field. INTRODUCTION In [2-4], the class of systems of random linear equations over a finite field or a ring R of cardinality q of the form } = °> / = , . . . , + J, (1) 7=1 is considered; here s is a fixed integer, a^

Journal

Discrete Mathematics and Applications – de Gruyter

Published: Jan 1, 1998

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