On the LP-convergence for multidimensional arrays of random variables

Let Z+, where d is an integer, denote the positive integer d-dimensional lattice points. The notation m≺ n, where m= (m1,m2, . . . ,md) and n= (n1,n2, . . . ,nd) ∈ Z+, means that mi ≤ ni, 1 ≤ i≤ d, |n| is used for ∏d i=1ni. Gut [2] proved that if {X ,Xn, n∈ Z+} is a d-dimensional array of i.i.d. random variables with E|X|p <∞ (0 < p < 2) and EX = 0 if 1 ≤ p < 2, then ∑ j≺nXj |n|1/p −→ 0 in L p as min 1≤i≤d ni −→∞, (1.1)