An Extreme-Value Analysis of the LIL for Brownian Motion

Abstract

We present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. Our result can be viewed as a new improvement to the LIL.

Darling--Erdős theorem for Lévy processes at zero

“…Einmahl and Mason [8] have obtained martingale Darling-Erdős theorems, and recently Dierickx and Einmahl [6] have established multivariate versions. Corresponding results for Brownian motion were established by Khoshnevisan et al [9].…”

A General Darling–Erdős Theorem in Euclidean Space

“…The authors asked whether this result can hold under the finite second moment assumption. Using Theorem 2.3 in combination with Theorem 1.1 in [13], we obtain the following general result:…”

“…In Sect. 6 we return to the real-valued case and show that if EX 2 I{|X| ≥ t} ∼ c(LLt) −1 that then (1.1) still remains valid if we replaceỸ byỸ − c. Finally, we answer a question which was posed in [13].…”

Darling–Erdős theorem for Lévy processes at zero

“…Einmahl and Mason (1989) have obtained martingale Darling-Erdős theorems, and recently Dierickx and Einmahl (2018) have established multivariate versions. Corresponding results for Brownian motion were established by Khoshnevisan et al (2005).…”