Complete convergence for weighted sums of negatively dependent random variables

A Strong Limit Theorem for Weighted Sums of Negatively Dependent Random Variables

“…(Sung, 2012). Let {X ni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise negatively dependent random variables with EX ni = 0, and {b n , n ≥ 1} a sequence of positive constants.…”

“…(Sung, 2012). Let {X ni , 1 ≤ i ≤ n, n ≥ 1} be an array of rowwise negatively dependent random variables such that EX ni = 0, and {X ni } is stochastically dominated by a random variable X satisfying E|X| p < ∞ for some p ≥ 1.…”

“…As corollaries, many results on the almost sure convergence and complete convergence for weighted sums of negatively dependent random variables are obtained. In particular, the results of Jing and Liang (2008), Sung (2012), and Wu (2010 can be obtained.…”

Strong laws of large numbers for arrays of row-wise extended negatively dependent random variables with applications

“…The previous statement extends Theorem 2.1 of [16] not only allowing p < 1 and enlarging the class of random triangular arrays (recall that arrays of row-wise negatively dependent random variables are arrays of row-wise END random variables with M n = 1 for all n) but also discarding its condition (2.4). In fact, supposing {X n,k , 1 k n, n 1} and {a n,k , 1 k n, n 1} as in Theorem 2.1 of [16], and c n,k = n 1/p a n,k in Corollary 1 we get that n k=1 a n,k X n,k converges completely to zero provided only max 1 k n |a n,k | = O n −1/p , n → ∞. Furthermore, Corollary 1 still improves assumption (4.11) and the moment condition presented in Corollary 4.4 of [15].…”

On complete convergence for Stout’s type weighted sums of NOD sequence

“…[9] is extended by many authors. Sung [10] and Cheng and Wang [3] extended Theorem 1.1 to random elements taking values in a Banach space, Wang and Su [14] to NA sequence, Cheng and Wang [4] to the products of the weighed sums of NA sequence, Wu [13] and Sung [11] to NOD sequence.…”