Results on the Sums of Associated Random Variables Motivated by the Parabolic Anderson Model.

摘要

We consider the parabolic Anderson model with white-noise potential and homogeneous initial condition. A characteristic feature of the large-time asymptotics of the solution field is that it exhibits intermittency, which is characterized by the occurrence of widely separated areas where almost all mass is concentrated.;Motivated by the definition of intermittency, our goal is to study limiting behavior of the total mass of the field of solutions {u( t, x) : x ∈ Zd } over boxes in Zd . As the first step towards our aim we derive limit distributions of sums of products of random exponentials, SN( n) = i=1Nn ebj =1nVij , where {Vij} are nonnegative, identically distributed, independent for each fixed i and associated for each fixed j random variables. Our choice for { Vij} is determined by an important property of the field {u(t, x) : x ∈ Zd } that the random variables in this field are associated. We investigate the influence of the growth rate of N(n) on the various limit laws that can arise for SN (n) when it is properly normalized and centered. We have determined two critical points, η1 < η 2, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. For N(n) < n exp(η2), we prove that the character of the limiting distribution is stable under a suitable normalization of SN (n).;Subsequently, we turn to another focus that can also help to reveal the occurrence of intermittency, namely the investigation of the influence of the maximum terms in the sum of associated random variables. We consider the sum of symmetric, jointly α-stable, identically distributed and associated random variables with distribution function that lies in the domain of attraction of an α-stable law. We show that the sum and the maximum summand are comparable in the sense that their ratio has a finite limiting distribution, which implies that the total mass is essentially dominated by its largest terms.