Note on Precise Rates in the Law of Iterated Logarithm for the Moment Convergence of I.I.D.: Random Variables under Sublinear Expectations

Abstract

Let X , X n , n ≥ 1 be a sequence of independent, identically distributed random variables under sublinear expectations with C V X 2 < ∞ , lim c ⟶ ∞ E X 2 − c + = 0 , and E ˘ X = E ˘ − X = 0 . Write S 0 = 0 , S n = ∑ k = 1 n X n , and M n = max 0 ≤ k ≤ n S k , n ≥ 1 . For d > 0 and a n = o log log n − d , we obtain the exact rates in the law of iterated logarithm of a kind of weighted infinite series of C V M n − ε + a n σ ¯ n log log n d + as ε ↓ 0 .