Random entire functions in linear dynamics

The rate of growth of an entire function f (z) = n=0 a n z n is given by the behaviour of max |z|r |f (z)| as r .Randomizing the coefficients, that is, considering f (z) = n=0 a n X n z n for a sequence (X n ) n of random variables, may decrease the rate of growth.This phenomenon was first observed by Paul Lvy in 1930, subsequently by Erds and Renyi, and more recently by the school of complex analysis around O. B. Skaskiv at Lviv.By answering a question of Skaskiv, we improve and generalize the known results.We also discuss an application to the dynamics of weighted shift operators on the space H(C) of entire functions.This is joint work with Kevin Agneessens.