On strong law of large numbers and growth rate for a class of random variables

Abstract

In this paper, we study the strong law of large numbers for a class of random variables satisfying the maximal moment inequality with exponent 2. Our results embrace the Kolmogorov strong law of large numbers and the Marcinkiewicz strong law of large numbers for this class of random variables. In addition, strong growth rate for weighted sums of this class of random variables is presented. MSC: 60F15

On Some Conditions for Strong Law of Large Numbers for Weighted Sums of END Random Variables under Sublinear Expectations

“…By the truncated random variable and some moment 2 Discrete Dynamics in Nature and Society inequalities, we testify that the SLLNs for the weighted sums hold for END random variables. In addition, our results extend the corresponding results of Shen et al [17] relative to the classical probability space. In the next part of this article, we provide some definitions under the sublinear expectations containing identical distribution, extended negatively dependent (END).…”

“…Proof of Theorem 7. When we replace {− ; ≥ 1} with { ; ≥ 1} in (16), we can obtain (17). So, we just need to prove (16).…”

On a General Approach to the Strong Laws of Large Numbers*

“…First they obtained Kolmogorov type maximal inequalities for moments. To this end, for NQD sequences, they applied Serfling's inequality (34). Then they derived Hájek-Rényi type inequalities for the probabilities and SLLN's similar to Theorem 4.…”

“…The conditions used in [38] are in terms of the covariances of the random variables. Applying Serfling's inequality (34), first a Kolmogorov type maximal inequality for the moments was obtained, then two general SLLN's were proved in [38]. The second SLLN rests on our Theorem 2.…”