ON THE RATE OF COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF ARRAYS OF RANDOM ELEMENTS

Abstract

Let {<TEX>$V_{nk},\;k\;{\geq}\;1,\;{\geq}\;1$</TEX>} be an array of rowwise independent random elements which are stochastically dominated by a random variable X with <TEX>$E\|X\|^{\frac{\alpha}{\gamma}+{\theta}}log^{\rho}(\|X\|)\;<\;{\infty}$</TEX> for some <TEX>${\rho}\;>\;0,\;{\alpha}\;>\;0,\;{\gamma}\;>\;0,\;{\theta}\;>\;0$</TEX> such that <TEX>${\theta}+{\alpha}/{\gamma}<2$</TEX>. Let {<TEX>$a_{nk},k{\geq}1,n{\geq}1$</TEX>) be an array of suitable constants. A complete convergence result is obtained for the weighted sums of the form <TEX>$\sum{^\infty_k_=_1}\;a_{nk}V_{nk}$</TEX>.