On a Weak Law of Large Numbers with Regularly Varying Normalizing Sequences

On weak laws of large numbers for maximal partial sums of pairwise independent random variables

“…irrespective of the joint distribution of the X n 's. Boukhari [4] presented an example showing that his result does not hold when p = 1. The proof of the sufficient part of Theorem 1 in [8] works well with pairwise independent random variables since we do not involve the maximal partial sums.…”

“…The above WLLN has been extended in several directions, see [12,13] for WLLNs with random indices for arrays of independent random variables taking values in Banach spaces, and see [4,6,9,10] and the references therein for WLLNs for dependent random variables and dependent random vectors. Boukhari [4,Theorem 1.2] showed that for 0…”

“…We postpone the proof of Theorem 2 to Section 2. From Theorem 3.2 of Boukhari [4], we have that if {X n , n ≥ 1} is a sequence of pairwise independent random variables, and {b n , n ≥ 1} is a sequence of positive constants, then…”