On the LP-convergence for multidimensional arrays of random variables

Abstract

For a d-dimensional array of random variables {Xn,n∈ℤ+d} such that {|Xn|p,n∈ℤ+d} is uniformly integrable for some 0<p<2, the Lp-convergence is established for the sums (1/|n|1/p) (∑j≺n(Xj−aj)), where aj=0 if 0<p<1, and aj=EXj if 1≤p<2.