On exact strong laws of large numbers under general dependence conditions

Abstract

We study the almost sure convergence of weighted sums of dependent random variables to a positive and finite constant, in the case when the random variables have either mean zero or no mean at all. These are not typical strong laws and they are called exact strong laws of large numbers. We do not assume any particular type of dependence and furthermore consider sequences which are not necessarily identically distributed. The obtained results may be applied to sequences of negatively associated random variables.

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

“…By Theorem 4.11, A 1 tends to zero almost surely. By Lemma 3.4 of [6] and since (19) is satisfied, n P (R n > c n ) < ∞. Then, the first Borel-Cantelli Lemma ensures that A 2 → 0 almost surely as n → ∞.…”

“…Condition (19) and Kronecker's lemma lead to A 3 → 0 almost surely. By Lemma 4.5 of [6] we have that lim n→∞ ER n I(R n ≤ c n ) µ(c n ) = 1…”

“…Proof. For the proof, we check that the assumptions of Theorem 4.1 in [6] are satisfied. Since (R n ) n≥1 is a sequence of independent random variables, Assumption (1.2) is satisfied.…”

Exact weak laws of large numbers with applications to ratios of random variables

“…Introduction. Let (R n ) n∈N be a sequence of independent random variables with the same distribution as the random variable R. It is well known (see [1], [3] and [6] for details) that if ER = 0 or ER = ∞, there are no sequences (M n ) n∈N such that 1 Mn n k=1 R k → 1 almost surely as n → ∞. Therefore it is a natural problem to find sequences (a n ) n∈N and (b n ) n∈N of real numbers such that 1 bn n k=1 a k R k → 1 almost surely as n → ∞.…”

“…Theorems of this kind are called exact strong law of large numbers, or weak exact laws of large numbers if we consider convergence in probability instead of almost sure convergence. We refer the reader to [1], [3] and [6] for details and further references on this topic.…”