## The Annals of Statistics

### Weak convergence of the sequential empirical processes of residuals in nonstationary autoregressive models

Shiqing Ling

#### Abstract

This paper establishes the weak convergence of the sequential empirical process $\hat{K}_n$ of the estimated residuals in nonstationary autoregressive models. Under some regular conditions, it is shown that $\hat{K}_n$ converges weakly to a Kiefer process when the characteristic polynomial does not include the unit root 1; otherwise $\hat{K}_n$ converges weakly to a Kiefer process plus a functional of stochastic integrals in terms of the standard Brownian motion. The latter differs not only from that given by Koul and Levental for an explosive AR(1) model but also from that given by Bai for a stationary ARMA model.

#### Article information

Source
Ann. Statist., Volume 26, Number 2 (1998), 741-754.

Dates
First available in Project Euclid: 31 July 2002

https://projecteuclid.org/euclid.aos/1028144857

Digital Object Identifier
doi:10.1214/aos/1028144857

Mathematical Reviews number (MathSciNet)
MR1626028

Zentralblatt MATH identifier
0932.62064

#### Citation

Ling, Shiqing. Weak convergence of the sequential empirical processes of residuals in nonstationary autoregressive models. Ann. Statist. 26 (1998), no. 2, 741--754. doi:10.1214/aos/1028144857. https://projecteuclid.org/euclid.aos/1028144857

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