Iterated Law of Iterated Logarithm

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...

A characterization of Chover-type law of iterated logarithm

ABSTRACT Let 0 < α ≤ 2 and - ∞ <β <∞. Let {X n ;n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set S n = X 1+⋯+X n , n ≥ 1. We say X satisfies the (α,β)-Chover-type law of the iterated logarithm (and write X∈C T L I L(α,β)) if [Formula: see text] almost surely. This paper is devoted to a characterization of X ∈C T L I L(α,β). We obtain sets of necessary and su...

An Analogue of the Law of the Iterated Logarithm

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...

Law of the Iterated Logarithm for Stationary Processes

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes · · ·X−1, X0, X1, · · · whose partial sums Sn = X1 + · · · + Xn are of the form Sn = Mn+Rn, where Mn is a square integrable martingale with stationary increments and Rn is a remainder term for which E(R 2 n) = o(n). Here we explore the Law of the Iterated Logarithm (LIL) for the...