On the rates of the other law of the logarithm∗
نویسنده
چکیده
Let X, X1, X2, . . . be i.i.d. random variables, and set Sn = X1 + . . . + Xn, Mn = maxk≤n |Sk|, n ≥ 1. Let an = o( √ n/ log n). By using the strong approximation, we prove that: if EX = 0, VarX = σ > 0 and E|X|2+2 < ∞ for some 2 > 0, then for any r > 1, lim 2↗1/√r−1 [2−2 − (r − 1)] ∞ ∑ n=1 nr−2P { Mn ≤ 2σ √ π2n/(8 log n) + an } = 4 π . We also show that the widest an is o( √ n/ log n).
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تاریخ انتشار 2004