Large deviation principles and loglog laws for observation processes

Abstract

Let (Xn, n ≥ 1) be an i.i.d. sequence of positive random variables with distribution function H. Let φH:={(n, Xn), n ≥ 1) be the associated observation process. We view φh as a measure on E:= [0, ∞) ∞ (0, φ] where φH (A) is the number of points of φH which lie in A. A family (Vs, s> 0) of transformations is defined on E in such a way that for suitable H the distributions of (VsφH, S > 0) satisfy a large deviation principle and that a related Strassen‐type law of the iterated logarithm also holds. Some consequent large deviation principles and loglog laws are derived for extreme values. Similar results are proved for φH replaced by certain planar Poisson processes.