Extreme value theory for space-time with random coefficients

Abstract

In this paper, we study the extreme value behavior of the space-time process given by X i ( t ) = ∑ j ≥ 1 Ψ ij ( t ) Z i − j ( t ) , t ∈ [ 0 , 1 ] , i ∈ Z . We assume that { Z i ( t ) } t ∈ [ 0 , 1 ] , i ∈ Z is a sequence of i.i.d random fields on [ 0 , 1 ] with values in the Skorokhod space D [ 0 , 1 ] of càdlàg functions (i.e., right-continuous functions with left limits) D [ 0 , 1 ] equipped with the J 1 topology. The coefficients { Ψ ij ( t ) } t ∈ [ 0 , 1 ] , i ∈ Z are processes with continuous sample paths. Our first aim is to establish a limit theory for point processes based on X(t). Secondly, using point processes, we study the limiting distribution of the normalized maximum process { a n − 1 max 1 ≤ i ≤ n X i ( t ) } t ∈ [ 0 , 1 ] . The result obtained in the second step can be viewed as extension of Balan who postponed the study of the behavior of maxima. It can also be considered as a generalization of Davis and Mikosch from deterministic real coefficients to random coefficients ( Ψ ij ) i ≥ 1 .