## The Annals of Probability

### Some Results on Increments of the Wiener Process with Applications to Lag Sums of I.I.D. Random Variables

#### Abstract

Let $W(t)$ be a standardized Wiener process. In this paper we prove that $\lim \sup_{T\rightarrow\infty} \max_{a_T \leq t \leq T}\frac{|W(T) - W(T - t)|}{\{2t\lbrack\log(T/t) + \log \log t\rbrack\}^{1/2}} = 1 \text{a.s.}$ under suitable conditions on $a_T$. In addition we prove various other related results all of which are related to earlier work by Csorgo and Revesz. Let $\{X_k\}$ be an i.i.d. sequence of random variables and let $S_N = X_1 + \cdots + X_N$. Our original objective was to obtain results similar to the ones obtained for the Wiener process but with $N$ replacing $T$ and $S_N$ replacing $W(T)$. Using the work of Komlos, Major, and Tusnady on the invariance principle, we obtain the desired results for i.i.d. sequences as immediate corollaries to our work for the Wiener process.

#### Article information

Source
Ann. Probab., Volume 11, Number 3 (1983), 609-623.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993505

Digital Object Identifier
doi:10.1214/aop/1176993505

Mathematical Reviews number (MathSciNet)
MR704547

Zentralblatt MATH identifier
0519.60030

JSTOR