Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes

Abstract

AbstractWe consider limit distributions of extremes of a process {Yn} satisfying the stochastic difference equation Yn-AnYn−1+Bn, n⩾1,Y0⩾0, where {An, Bn} are i.i.d. R2+-valued random pairs, A special case of interest is when {Yn} is derived from a first order ARCH process. Parameters of the limit law are exhibited; some are hard to calculate explicitly but easy to simulate