A conditional dichotomy theorem for stochastic processes with independent increments

Abstract

Let X( t) and Y( t) be two stochastically continuous processes with independent increments over [0, T] and Lévy spectral measures M t and N t , respectively, and let the “time-jump” measures M and N be defined over [0, T] × R ⧹{0} by M(( t 1, t 2] × A) = M t 2 ( A) − M t 1 ( A) and N(( T 1, t 2] × A) = N t 2 ( A) − N t 1 ( A). Under the assumption that M is equivalent to N, it is shown that the measures induced on function space by X( t) and Y( t) are either equivalent or orthogonal, and necessary and sufficient conditions for equivalence are given. As a corollary a complete characterization of the set of admissible translates of such processes is obtained: a function f is an admissible translate for X( t) if and only if it is an admissible translate for the Gaussian component of X( t). In particular, if X( t) has no Gaussian component, then every nontrivial translate of X( t) is orthogonal to it.