## Illinois Journal of Mathematics

- Illinois J. Math.
- Volume 45, Number 2 (2001), 559-580.

### Pair correlations and $U$-statistics for independent and weakly dependent random variables

István Berkes, Walter Philipp, and Robert Tichy

#### Abstract

We prove a Glivenko-Cantelli type strong law of large numbers for the pair correlation of independent random variables. Except for a few powers of logarithms the results obtained are sharp. Similar estimates hold for the pair correlation of lacunary sequences $\{n_k\omega\}\mod 1$.

#### Article information

**Source**

Illinois J. Math., Volume 45, Number 2 (2001), 559-580.

**Dates**

First available in Project Euclid: 13 November 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1258138356

**Digital Object Identifier**

doi:10.1215/ijm/1258138356

**Mathematical Reviews number (MathSciNet)**

MR1878619

**Zentralblatt MATH identifier**

1015.11034

**Subjects**

Primary: 60F15: Strong theorems

Secondary: 11K06: General theory of distribution modulo 1 [See also 11J71]

#### Citation

Berkes, István; Philipp, Walter; Tichy, Robert. Pair correlations and $U$-statistics for independent and weakly dependent random variables. Illinois J. Math. 45 (2001), no. 2, 559--580. doi:10.1215/ijm/1258138356. https://projecteuclid.org/euclid.ijm/1258138356