摘要
Let {X, X<sub>n</sub>; n≥1} be i.i.d.r.v.’s taking values in a separable Banach space （B,||·||）such that EX=0 and Ef<sup>2</sup>（X）【+∞, ?∈6B<sup>*</sup>, and S<sub>n</sub>=X<sub>1</sub>+…+X<sub>n</sub> for n≥1. The purposeof this paper is to study the rates of convergence to zero of P(inf||Sn/（2nloglogn）<sup>1/2</sup>-x||≥ε) and P(sup inf||S<sub>k</sub>/（2kloglogk）<sup>1/2</sup>-x||≥ε) （?ε】0） under precisely necessary and sufficientconditions. We also give new necessary and sufficient conditions for X to satisfy the boundand compact law of the iterated logarithm, respectively. Our results improve some resultsof Darling and Robbins （1967） as well as Davis （1968） even in the case B=R.
Let {X, X_n; n≥1} be i.i.d.r.v.'s taking values in a separable Banach space (B,||·||)such that EX=0 and Ef^2(X)<+∞, ?∈6B~*, and S_n=X_1+…+X_n for n≥1. The purposeof this paper is to study the rates of convergence to zero of P(inf||Sn/(2nloglogn)^(1/2)-x||≥ε) and P(sup inf||S_k/(2kloglogk)^(1/2)-x||≥ε) (?ε>0) under precisely necessary and sufficientconditions. We also give new necessary and sufficient conditions for X to satisfy the boundand compact law of the iterated logarithm, respectively. Our results improve some resultsof Darling and Robbins (1967) as well as Davis (1968) even in the case B=R.
基金
Project supported by the National Natural Science Foundation of China.
