Limit distribution of the coefficients of polynomials with only unit roots1. Issue 4 (11th October 2013)

• Record Type: Journal Article
• Authors: Hwang, Hsien‐Kuei; Zacharovas, Vytas
• Is Part Of: Random structures & algorithms. Volume 46:Issue 4(2015)
• Page Start: 707
• Page End: 738
• Publication Date: 2013-10-11
• Subjects: Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20516
• Languages: English
• ISSNs: 1042-9832

Abstract

<abstract abstract-type="main"> <title>Abstract</title> <p>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</p> </abstract>