## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 1 (1974), 147-154.

### Weak Convergence of Multidimensional Empirical Processes for Stationary $\phi$-Mixing Processes

#### Abstract

For a stationary $\phi$-mixing sequence of stochastic $p(\geqq 1)$-vectors, weak convergence of the empirical process (in the $J_1$-topology on $D^p\lbrack 0, 1 \rbrack)$ to an appropriate Gaussian process is established under a simple condition on the mixing constants $\{\phi_n\}$. Weak convergence for random number of stochastic vectors is also studied. Tail probability inequalities for Kolmogorov Smirnov statistics are provided.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 1 (1974), 147-154.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996760

**Digital Object Identifier**

doi:10.1214/aop/1176996760

**Mathematical Reviews number (MathSciNet)**

MR402845

**Zentralblatt MATH identifier**

0276.60030

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

Secondary: 60B10: Convergence of probability measures

**Keywords**

$D^p \lbrack 0, 1 \rbrack$ space empirical processes Gaussian process random sample size Skorokhod $J_1$-topology tightness weak convergence

#### Citation

Sen, Pranab Kumar. Weak Convergence of Multidimensional Empirical Processes for Stationary $\phi$-Mixing Processes. Ann. Probab. 2 (1974), no. 1, 147--154. doi:10.1214/aop/1176996760. https://projecteuclid.org/euclid.aop/1176996760