## The Annals of Probability

- Ann. Probab.
- Volume 4, Number 3 (1976), 470-473.

### On Weak Convergence of Extremal Processes

#### Abstract

Lamperti in 1964 showed that the convergence of the marginals of an extremal process generated by independent and identically distributed random variables implies the full weak convergence in the Skorohod $J_1$-topology. This result is generalized to the $k$th extremal process and to random variables which need not be identically distributed. The proof here is based on the weak convergence of a certain point-process (which counts the number of up-crossings of the variables) to a two-dimensional nonhomogeneous Poisson process.

#### Article information

**Source**

Ann. Probab., Volume 4, Number 3 (1976), 470-473.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996096

**Digital Object Identifier**

doi:10.1214/aop/1176996096

**Mathematical Reviews number (MathSciNet)**

MR400330

**Zentralblatt MATH identifier**

0401.60058

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60B10: Convergence of probability measures

Secondary: 60G99: None of the above, but in this section

**Keywords**

Extremal processes multivariate $k$-dimensional extremal processes nonhomogeneous two-dimensional Poisson process $D\lbrack a, b \rbrack$ space Skorohod space of functions with several parameters weak convergence

#### Citation

Weissman, Ishay. On Weak Convergence of Extremal Processes. Ann. Probab. 4 (1976), no. 3, 470--473. doi:10.1214/aop/1176996096. https://projecteuclid.org/euclid.aop/1176996096