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Asymptotic results for random polynomials on the unit circle

Tucci G. H., Whiting P.

Probability and Mathematical Statistics

2014

Vol. 34, Fasc. 2

181--197

In this paper we study the asymptotic behavior of the maximum magnitude of a complex random polynomial with i.i.d. uniformly distributed random roots on the unit circle. More specifically, let {nk}∞k=1 be an infinite sequence of positive integers and let {zk}∞k=1 be a sequence of i.i.d. uniformly distributed random variables on the unit circle. The above pair of sequences determine a sequence of random polynomials PN(z) = ΠNk=1 (z − zk)nk with random roots on the unit circle and their corresponding multiplicities. In this work, we show that subject to a certain regularity condition on the sequence {nk}∞k=1, the log maximum magnitude of these polynomials scales as sNI*, where s2N = ΣNk=1 n2k and I* is a strictly positive random variable.