Limit Theorems for Variational Sums

Abstract

Limit theorems in the sense of a.s. convergence, convergence in ${L_1}$-norm and convergence in distribution are proved for variational series. In the first two cases, if g is a bounded, nonnegative continuous function satisfying an additional assumption at zero, and if $\{ X(t),0 \leq t \leq T\}$ is a stochastically continuous stochastic process with independent increments, with no Gaussian component and whose trend term is of bounded variation, then the sequence of variational sums of the form $\Sigma _{k = 1}^ng(X({t_{nk}}) - X({t_{n,k - 1}}))$ is shown to converge with probability one and in ${L_1}$-norm. Also, under the basic assumption that the distribution of the centered sum of independent random variables from an infinitesimal system converges to a (necessarily) infinitely divisible limit distribution, necessary and sufficient conditions are obtained for the joint distribution of the appropriately centered sums of the positive parts and of the negative parts of these random variables to converge to a bivariate infinitely divisible distribution. A characterization of all such limit distributions is obtained. An application is made of this result, using the first theorem, to stochastic processes with (not necessarily stationary) independent increments and with a Gaussian component.