## Bernoulli

- Bernoulli
- Volume 9, Number 3 (2003), 497-515.

### Compound Poisson limit theorems for high-level exceedances of some non-stationary processes

Lise Bellanger and Gonzalo Perera

#### Abstract

We show the convergence to a compound Poisson process of the high-level exceedances point process $N_n(B)= \sum_{j/n\in B} 1_{\{X_j>u_n\}}$, where $X_n=\varphi(\xi_n,Y_n) $, $ \varphi $ is a (regular) regression function, $u_n$ grows to infinity with $n$ in some suitable way, $\xi$ and $Y$ are mutually independent, $\xi$ is stationary and weakly dependent, and $Y$ is non-stationary, satisfying some ergodic conditions. The basic technique is the study of high-level exceedances of stationary processes over suitable collections of random sets.

#### Article information

**Source**

Bernoulli, Volume 9, Number 3 (2003), 497-515.

**Dates**

First available in Project Euclid: 6 October 2003

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1065444815

**Digital Object Identifier**

doi:10.3150/bj/1065444815

**Mathematical Reviews number (MathSciNet)**

MR1997494

**Zentralblatt MATH identifier**

1049.60043

**Keywords**

asymptotically ponderable collections of sets compound Poisson process convergence exceedances level sets mean occupation measures point processes

#### Citation

Bellanger, Lise; Perera, Gonzalo. Compound Poisson limit theorems for high-level exceedances of some non-stationary processes. Bernoulli 9 (2003), no. 3, 497--515. doi:10.3150/bj/1065444815. https://projecteuclid.org/euclid.bj/1065444815