The Annals of Statistics

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

Shiqing Ling

Full-text: Open access


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

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

First available in Project Euclid: 31 July 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G30: Order statistics; empirical distribution functions 60F17: Functional limit theorems; invariance principles
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62F05: Asymptotic properties of tests

Brownian motions Kiefer process sequential empirical processes nonstationary autoregressive model weak convergence


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.

Export citation


  • BAI, J. 1993. On the partial sums of residuals in autoregressive and moving average models. J. Time. Ser. Anal. 14 247 260. Z.
  • BAI, J. 1994. Weak convergence of the sequential empirical processes of residuals in ARMA models. Ann. Statist. 22 2051 2061. Z.
  • BICKEL, P. J. and WICHURA, M. J. 1971. Convergence for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 1656 1670. Z.
  • BILLINGSLEY, P. 1968. Convergence of Probability Measures. Wiley, New York. Z.
  • BOLDIN, M. V. 1982. Estimation of the distribution of noise in an autoregressive scheme. Theory Probab. Appl. 27 866 871. Z.
  • CHAN, N. H. and WEI, C. Z. 1988. Limiting distributions of least squares estimates of unstable autoregressive processes. Ann. Statist. 16 367 401. Z.
  • CHUNG, K. L. 1968. A Course in Probability Theory. Harcourt Brace and World, New York. Z.
  • HALL, P. and HEy DE, C. C. 1980. Martingale Limit Theory and Its Applications. Academic Press, San Diego. Z.
  • JEGANATHAN, P. 1991. On the asy mptotic behavior of least squares estimators in AR time series with roots near the unit circle. Econometric Theory 7 269 306. Z.
  • KOUL, H. L. 1991. A weak convergence result useful in robust autoregression. J. Statist. Plann. Inference 29 1291 1308. Z.
  • KOUL, H. L. and LEVENTAL, S. 1989. Weak convergence of the residual empirical process in explosive autoregression. Ann. Statist. 17 1784 1794. Z.
  • KREISS, P. 1991. Estimation of the distribution of noise in stationary processes. Metrika 38 285 297. Z.
  • LEE, S. 1991. Testing whether a time series is Gaussian. Ph.D. dissertation, Dept. Mathematics, Univ. Mary land. Z.
  • SHORACK, G. R. and WELLNER, J. A. 1986. Empirial Processes with Applications to Statistics. Wiley, New York. Z.
  • STRAF, M. J. 1970. Weak convergence of stochastic processes with several parameters. Proc. Fourth Berkeley Sy mp. Math. Statist. Probab. 187 221. Univ. California Press, Berkeley. DEPARTMENT OF STATISTICS DEPARTMENT OF ECONOMICS UNIVERSITY OF HONG KONG UNIVERSITY OF WESTERN AUSTRALIA