The Law of the Iterated Logarithm in Analaysis

The Law of the Iterated Logarithm in Analaysis

In this paper, we first discuss the history of the law of the iterated logarithm. We then focus our discussion on how it was introduced in analysis. Finally we mention different types of law of the iterated logarithm and state some of the recent developments. In order to discuss the history and developments of law of the iterated logarithm, some definitions and theorems are in order: Definition...

On the law of the iterated logarithm.

The law of the iterated logarithm provides a family of bounds all of the same order such that with probability one only finitely many partial sums of a sequence of independent and identically distributed random variables exceed some members of the family, while for others infinitely many do so. In the former case, the total number of such excesses has therefore a proper probability distribution...

Precise Asymptotics in Chung’s law of the iterated logarithm∗

Let X, X1, X2, . . . be i.i.d. random variables with mean zero and positive, finite variance σ2, and set Sn = X1 + . . . + Xn, n ≥ 1. We prove that, if EX2I{|X| ≥ t} = o((log log t)−1) as t →∞, then for any a > −1 and b > −1, lim 2↗1/√1+a ( 1 √ 1+a − 2)b+1 ∞n=1 (log n) a(log log n)b n P { maxk≤n |Sk| ≤ √ σ2π2n 8 log log n(2 + an) } = 4 π ( 1 2(1+a)3/2 )b+1Γ(b + 1), whenever an = o(1/ log log n).

Exponential Bounds in the Law of Iterated Logarithm for Martingales

In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norm-ing in the spirit of the classical Law of Iterated Logarithm.

An Analogue of the Law of the Iterated Logarithm