On The Properties Of Random Entire Functions#

Abstract

We consider random entire functions of the form where (X j (ω), j≥0) is a sequence of independent and identically distributed random variables defined on the probability space (Ω, A, P), ω∈Ω and z=re iθ . An important characteristic of f(z, ω), is sup θ∈[−π,π]| f(re iθ ,ω)|, with its growth as r→∞ of particular significance. Results describing the latter using the Wiman–Valiron method are known for random entire functions generated by bounded random variables. In contrast to this work, we consider the important class of Gaussian entire functions and determine the properties of sup θ∈I ε | f(re iθ ,ω)| for large r, when I ε⊂[−π,π] excludes a neighbourhood of θ=0. As our techniques exploit the assumption that f(re iθ ,ω),θ∈[−π,π] is a Gaussian process, we show that a wide range of random entire functions have Gaussian characteristics for sufficiently large r. The latter may be of independent interest as it can be used to determine the expected number of zero crossings of the real and imaginary parts of f(re iθ ,ω) and related quantities. #This work was carried out as part of Technology Group 10 of the MoD Corporate Research Programme. (©British Crown Copyright 2000/DERA.)