BROWNIAN COVARIANCE AND CENTRAL LIMIT THEOREM FOR STATIONARY SEQUENCES

In this paper, using a Brownian motion we define a new type of covariance for random vectors with a finite second moment. The advantage of Brownian covariance is in the fact that it is equal to zero if and only if the random vectors are independent. We can apply Brownian covariance to random vectors, generally speaking, of different sizes if it is invariant with respect to rotations of axes where the data are given. If, in the definition of Brownian covariance, we replace Brownian motion by an identical function, then we obtain a modulus of the classical Pearson covariance coefficient, and therefore, Brownian covariance generalizes the classical. Brownian covariance is applied to the proof of necessary and sufficient conditions for the validity of the central limit theorem for random sequences which are stationary in a narrow sense.