## The Annals of Probability

- Ann. Probab.
- Volume 9, Number 6 (1981), 988-996.

### Improved Erdos-Renyi and Strong Approximation Laws for Increments of Partial Sums

#### Abstract

Let $X_1, X_2,\cdots$ be an i.i.d. sequence with $EX_1 = 0, EX^2_1 = 1, Ee^{tX_1} < \infty (|t| < t_0)$, and partial sums $S_n = X_1 + \cdots + X_n$. Starting from some analogous results for the Wiener process, this paper studies the almost sure limiting behaviour of $\max_{0 \leq n \leq N - a_N} a^{-1/2}_N (S_{n + a_N} - S_n)$ as $N \rightarrow \infty$ under various conditions on the integer sequence $a_N$. Improvements of the Erdos-Renyi law of large numbers for partial sums are obtained as well as strong invariance principle-type versions via the Komlos-Major-Tusnady approximation. An appearing gap between these two results is also going to be closed.

#### Article information

**Source**

Ann. Probab., Volume 9, Number 6 (1981), 988-996.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176994269

**Digital Object Identifier**

doi:10.1214/aop/1176994269

**Mathematical Reviews number (MathSciNet)**

MR632971

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 60F10: Large deviations 60F17: Functional limit theorems; invariance principles 60G15: Gaussian processes 60G17: Sample path properties

**Keywords**

Increments of partial sums strong approximations strong invariance principles large deviations

#### Citation

Csorgo, M.; Steinebach, J. Improved Erdos-Renyi and Strong Approximation Laws for Increments of Partial Sums. Ann. Probab. 9 (1981), no. 6, 988--996. doi:10.1214/aop/1176994269. https://projecteuclid.org/euclid.aop/1176994269