Some limit theorems for weighted negative quadrant dependent random variables with infinite mean

Abstract

In the present paper, we will investigate weak laws of large numbers for weighted pairwise NQD random variables with infinite mean. The almost sure upper and lower bounds for a particular normalized weighted sum of pairwise NQD nonnegative random variables are established also.

On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean

“…and the result follows immediately from (16). The second part of the Theorem follows easily since, by the first part it suffices to prove the same relation with Y n and Y m in place of R n and R m respectively.…”

“…Observe that by condition (16) EY m = ∞ and by employing condition (16) again it can be easily verified that…”

“…Remark 3.10 In both papers [19] and [16] (see Corollary 2.1 in both), it is proven that in the case where α ∈ (0, 1) the limit of the weighted partial sum is equal to zero, i.e. the weak law is established but it is not an exact weak law.…”

“…Remark 3.2 Theorem 3.1 has been recently generalized in [16] to the case of negative quadrant dependent random variables.…”

“…Assumption (1.3) is satisfied by Theorem 4.1 (b). Let m be the integer considered in Theorem 4.1 (b); then the assumption (16) ensures that…”