Asymptotic property for some series of probability

Convergence rates in precise asymptotics for a kind of complete moment convergence

“…We first prove the part (a) of this theorem. Recall that He and Xie [14] obtained that Hence the part (a) holds. Next, we prove the part (b).…”

“…We first prove the part (a) of this theorem. Recall that He and Xie [14] obtained that ∞ n=1 P(|S n | ≥ ǫn) = ǫ −2 (σ 2 + o(ǫ q−2 )). Hence the part (a) holds.…”

“…This topic has been studied extensively. For example, Klesov [17] proved that if E[X] = 0, E[X 2 ] > 0 and E[|X| 3 ] < ∞, then [17] with E[|X| q ] < ∞ for some q ∈ (2, 3], He and Xie [14] got a much faster rate for Heyde's result. Recently, Gut and Steinebach [12,13], and Kong [18] also extended the Klesov's [17] result, respectively, and got some new results.…”

Precise Asymptotics for Complete Integral Convergence under Sublinear Expectations

“…Many related results have been obtained in the probabilistic space. eir recent results can be found in the studies of Heyde [19]; Liu and Lin [20]; Zhao [21]; Li [22]; Zhou [23]; Gut and Steinebach [24]; He and Xie [25]; Wang et al [26][27][28][29]; Spaataru [30]; and Kong and Dai [31]. However, in sublinear expectations, due to the uncertainty of expectation and capacity, the precise asymptotics is essentially different from the ordinary probability space.…”