Strong laws of large numbers for arrays of rowwise independent random elements

Abstract

Let {Xnk} be an array of rowwise independent random elements in a separable Banach space of type p+δ with EXnk=0 for all k, n. The complete convergence (and hence almost sure convergence) of n−1/p∑k=1nXnk to 0, 1≤p<2, is obtained when {Xnk} are uniformly bounded by a random variable X with E|X|2p<∞. When the array {Xnk} consists of i.i.d, random elements, then it is shown that n−1/p∑k=1nXnk converges completely to 0 if and only if E‖X11‖2p<∞.

Keywords

• random elements
• strong laws of large numbers
• complete convergence
• Rademacher type p+&#948; spaces.