Let W(t) denote a standard Wiener process for 0 ≤ t < ∞. We compute the probability that W(t) ≥ t½ A(t) for some t ≥ 1 (or for some t ≥ 0) for a certain class of functions A(t), including functions which are ∼ (2 log log t)½ as t → ∞. We also give an invariance principle which states that this probability is the limit as m → ∞ of the probability that sn ≥ n½A(n/m) for some n ≥ m (or for some n ≥ 1), where sn is the sum of n independent and identically distributed random variables with mean 0 and variance 1.
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机译：令W（t）表示0≤t <∞的标准Wiener过程。对于某类函数A（t），我们计算W（t）≥t 1/2 A（t）的概率，对于某些类的函数A（t），包括是〜（2 log log t） 1/2 为t→∞。我们还给出了不变性原则，该原则规定此概率是sn≥ n ½ A （< em> n / m ）用于某些 n ≥ m （或用于某些 n ≥1），其中 sn 是 n 个独立且均匀分布的随机变量的总和，均值0，方差1。
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