A note on strong convergence of sums of dependent random variables.
The goal of this paper is to show that, in most strong laws of large numbers, the nth partial sum can be replaced with the largest of the first n sums. Moreover it is shown that the usual assumptions of independence and common distribution are unnecessary and that these results apply also to strong laws for Banach valued random elements.
The strong consistency of least squares estimates in multiples regression models with i.i.d. errors is obtained under assumptions on the design matrix and moment restrictions on the errors.
In this paper we obtain a strong invariance principle for negatively associated random fields, under the assumptions that the field has a finite th moment and the covariance coefficient exponentially decreases to . The main tools are the Berkes-Morrow multi-parameter blocking technique and the Csörgő-Révész quantile transform method.
By the method of Rio [10], for a locally square integrable periodic function f, we prove for almost every x and t > 0.