Hole probability for entire functions represented by Gaussian Taylor series

Research output: Contribution to journalArticlepeer-review

Abstract

Consider the Gaussian entire function f(z) = Σn=0ζnanzn, where {ζn} is a sequence of independent and identically distributed standard complex Gaussians and {an} is some sequence of non-negative coefficients, with a0 > 0. We study the asymptotics (for large values of r) of the hole probability for f (z), that is, the probability PH(r) that f(z) hcisk {{pipe}z{pipe} < r}. We prove that log PH(r) = -S(r) + o(S(r)), where S(r) = 2·Σn≥0log+(anrn) as r tends to ∞ outside a deterministic exceptional set of finite logarithmic measure.

Original languageEnglish
Pages (from-to)493-507
Number of pages15
JournalJournal d'Analyse Mathematique
Volume118
Issue number2
DOIs
StatePublished - Nov 2012

Fingerprint

Dive into the research topics of 'Hole probability for entire functions represented by Gaussian Taylor series'. Together they form a unique fingerprint.

Cite this