Topics in the Value Distribution of Random Analytic Functions

Abstract

This thesis is concerned with the behavior of random analytic functions. In particular, we are interested in the value distribution of Taylor series with independent random coefficients. We begin with a study of the properties of Fourier series with random signs. The main result states that the logarithm of such series is integrable (to any power). Using this result, we answer an old question of J.-P. Kahane, concerning the range of random Taylor series in the unit disk. In addition, we prove a law of large numbers for the number of zeros of entire functions given by Taylor series with random signs. Then we examine some 'rare' events related to the zero set of Gaussian entire functions (given by a Taylor series). In particular, we are interested in the 'hole' event, where the function has no zeros inside a large disk, centered at the origin. We give precise logarithmic asymptotics for the probability of this event, as the radius tends to infinite, depending on the variance of the coefficients of the series. It should be mentioned that our results do not assume any 'regularity' conditions.