The Annals of Probability

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

D. L. Hanson and Ralph P. Russo

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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

Permanent link to this document
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
links.jstor.org

Subjects
Primary: 60F15: Strong theorems
Secondary: 60G15: Gaussian processes 60G17: Sample path properties

Keywords
Increments of a Wiener process Wiener process law of iterated logarithm lag sums sums of random variables

Citation

Hanson, D. L.; Russo, Ralph P. Some Results on Increments of the Wiener Process with Applications to Lag Sums of I.I.D. Random Variables. Ann. Probab. 11 (1983), no. 3, 609--623. doi:10.1214/aop/1176993505. https://projecteuclid.org/euclid.aop/1176993505


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