Complete convergence for widely acceptable random variables under the sublinear expectations

Precise Asymptotics in the Law of the Iterated Logarithm under Sublinear Expectations

“…x/y . (16) Combining (16) with the inequality derived from it with −X and −X k in place of X and X k , respectively, leads to (5). □ Remark 1.…”

“…Chen [13] proved strong laws of large numbers for sublinear expectations. For more results about limit theorems under sublinear expectations, the interested reader could refer to the studies of Hu et al [14], Fang et al [15], Kuczmaszewska [16], Wang and Wu [17], Hu and Yang [18], Zhang [19], and references therein.…”

Complete Convergence for Weighted Sums of Widely Acceptable Random Variables under Sublinear Expectations

“…Definition 5 (widely acceptable, see [19]). Let Y n , n ≥ 1 be a sequence of random variables in a sublinear expectation space(Ω, H,Ê).…”

“…Proof. We can easily get ( 15) from eorem 1 of [19]. □ Lemma 2 Suppose X ∈ H, p > 0, for any constant c > 0:…”

“…And, similar to Case 1, we also get I 221 < ∞. Next, we discuss the situation of (A.1) or (A.2) holds.If (A.1) holds, according to(19), we can get…”