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
be an infinite sequence of positive integers and let
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
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
, the log maximum magnitude of these polynomials scales
as
, where
and
is a strictly positive random variable.
2000 AMS Mathematics Subject Classification: Primary: 60F99; Secondary:
60B10.
Keywords and phrases: Random polynomials, Brownian bridge, stochastic
process.