Some Functional Limit Theorems for Dependent Random Variables

Abstract

We prove theorems on weak convergence of random elements $X_m(\omega, t), 0 \leqq t \leqq 1$, to a Gaussian process. In Part I, these random elements are constructed on the basis of the linear means of a lacunary trigonometric series $\sum a_j \cos n_j\omega$. In Part II, the lacunarity hypothesis is dropped and replaced by the hypothesis of linear independence of the real numbers $n_j$.