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Properties of Macroscopic Quantum Effects and Dynamic Natures of Electrons in Superconductors
Precise rates in the law of logarithm for i.i.d. random variables
AbstractLet {X, Xn; n ≥ 1} be a sequence of i.i.d. random variables. Set Sn = X1 + X2 + … + Xn and Mn = maxk≤n |Sk|, n ≥ 1. By using the strong approximation method, we obtain that for any −1 < b ≤ 1, limε↘0ε2b+2∑n=1∞(logn)bnP(Mn≥εσnlogn)=2E|N|(2b+2)b+1∑k=0∞(−1)k(2k+1)2b+2 if and only if Ex = 0 and Ex2 < ∞, which strengthen and extend the result of Gut and Spǎtaru [1], where N is the standard normal random variable. Furthermore, L2 convergence and a.s. convergence are also discussed
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