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WROCŁAW UNIVERSITY
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TECHNOLOGY

Contents of PMS, Vol. 34, Fasc. 2,
pages 181 - 197
 

ASYMPTOTIC RESULTS FOR RANDOM POLYNOMIALS ON THE UNIT CIRCLE

Gabriel H. Tucci
Philip Whiting

Abstract: 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 (n )∞ k k=1 be an infinite sequence of positive integers and let (z )∞ k 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 P (z) = ∏N (z - z )nk N k=1 k 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 (n )∞ k k=1 , the log maximum magnitude of these polynomials scales as * sN I , where 2 ∑N 2 sN = k=1 nk and * I is a strictly positive random variable.

2000 AMS Mathematics Subject Classification: Primary: 60F99; Secondary: 60B10.

Keywords and phrases: Random polynomials, Brownian bridge, stochastic process.

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