Abstract

Let (X<inf>nk</inf>), k = 1, ..., k<inf>n</inf> ; n = 1, 2, ... be a double sequence of infinitesimal random variables which are rowwise independent. In this paper, we give necessary and sufficient conditions for the sequence of distribution functions of S<inf>n</inf>⁽r ⁾ = <inf>(</inf>g<inf>(</inf>X<inf>n</inf><inf>1))</inf>^r +L + <inf>(</inf>g<inf>(</inf>X<inf>nkn))</inf>^r B<inf>n</inf> (r) to weakly converge to a limiting distribution function F<inf>r</inf> for each natural number r, and also for convergence of (F<inf>r</inf>).