Long- and short-range dependent sequences under exponential subordination

Abstract

Abstract We consider a strictly stationary long-range-dependent process ( Z i ) i =1 ∞ with standard exponential marginals and its subordinated process ( G ( Z i )) i =1 ∞ for some real function G . We prove that, analogously to the case of Gaussian subordination, the asymptotic behaviour of partial-sum process of a long-range-dependent sequence ( G ( Z i )) i =1 ∞ is the same as that of the first nonvanishing term of its Laguerre expansion ( L m ( Z i )) i =1 ∞ . Furthermore, convergence in distribution of partial-sum process to a linear combination of generalized Hermite processes of rank 2 m is shown. This leads to noncentral limit theorems for an empirical process and kernel density estimators. A parallel problem of convergence in distribution for partial sums of a short-range-dependent exponentially subordinated process to a constant multiple of Wiener process is solved for G having finite Laguerre expansion.