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Limit distribution of the coefficients of polynomials with only unit roots
Author(s) -
Hwang HsienKuei,
Zacharovas Vytas
Publication year - 2015
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20516
Subject(s) - mathematics , central limit theorem , illustration of the central limit theorem , limit (mathematics) , random variable , moment generating function , probability distribution , unit (ring theory) , moment (physics) , probability generating function , representation (politics) , variance (accounting) , unit circle , second moment of area , distribution (mathematics) , contrast (vision) , pure mathematics , combinatorics , sum of normally distributed random variables , convergence of random variables , mathematical analysis , statistics , computer science , law , geometry , business , accounting , classical mechanics , political science , physics , mathematics education , politics , artificial intelligence
Abstract We consider sequences of random variables whose probability generating functions have only roots on the unit circle, which has only been sporadically studied in the literature. We show that the random variables are asymptotically normally distributed if and only if the fourth central and normalized (by the standard deviation) moment tends to 3, in contrast to the common scenario for polynomials with only real roots for which a central limit theorem holds if and only if the variance is unbounded. We also derive a representation theorem for all possible limit laws and apply our results to many concrete examples in the literature, ranging from combinatorial structures to numerical analysis, and from probability to analysis of algorithms. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46,707–738, 2015