The Annals of Probability

Long-range dependence and Appell rank

Donatas Surgailis

Full-text: Open access


We study limit distributions of sums $S_N^{(G)} = \sum_{t=1}^N G(X_t)$ of nonlinear functions $G(x)$ in stationary variables of the form $X_t = Y_t + Z_t$, where ${Y_t}$ is a linear (moving average) sequence with long-range dependence, and ${Z_ t}$ is a (nonlinear) weakly dependent sequence. In particular, we consider the case when ${Y_ t}$ is Gaussian and either (1)${Z_t}$ is a weakly dependent multilinear form in Gaussian innovations, or (2) ${Z_t}$ is a finitely dependent functional in Gaussian innovations or (3)${Z_t}$ is weakly dependent and independent of $Y_t$ . We show in all three cases that the limit distribution of $S^(G)_N$ is determined by the Appell rank of $G( x)$, or the lowest $k\geq 0$ such that $a_k = \partial^k E\{G(X_0+c)\}/\partial c^k|_{c=0 \not= 0$.

Article information

Ann. Probab., Volume 28, Number 1 (2000), 478-497.

First available in Project Euclid: 18 April 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G15: Gaussian processes 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Long-range dependence noncentral limit theorems reduction principle Appell polynomials Appell rank


Surgailis, Donatas. Long-range dependence and Appell rank. Ann. Probab. 28 (2000), no. 1, 478--497. doi:10.1214/aop/1019160127.

Export citation


  • Avram,F. and Taqqu,M. S. (1987). Generalized powers of strongly dependent random variables. Ann. Probab. 15 767-775.
  • Beran,J. (1992). Statistical methods for data with long-range dependence (with discussions). Statist. Sci. 7 404-427.
  • Breuer,P. and Major,P. (1983). Central limit theorem for non-linear functionals of Gaussian fields. J. Multivariate Anal. 13 425-441.
  • Dobrushin,R. L. (1979). Gaussian and their subordinated self-similar random generalized fields. Ann. Probab. 7 1-28.
  • Dobrushin,R. L. and Major,P. (1979). Non-central limit theorems for non-linear functionals of Gaussian fields.Wahrsch. Verw. Gebiete 50 27-52.
  • Giraitis,L. and Surgailis,D. (1985). CLT and other limit theorems for functionals of Gaussian processes.Wahrsch. Verw. Gebiete 70 191-212.
  • Giraitis,L. and Surgailis,D. (1986). Multivariate Appell polynomials and the central limit theorem. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 21-71. Birkh¨auser, Boston.
  • Giraitis,L. and Surgailis,D. (1989). A limit theorem for polynomials of linear process with long range dependence. Lithuanian Math. J. 29 290-311.
  • Giraitis,L. and Taqqu,M. S. (1997). Limit theorems for bivariate Appell polynomials I: Central limit theorems. Probab. Theory Related Fields 107 359-381.
  • Giraitis,L.,Taqqu,M. S. and Terrin,N. (1998). Limit theorems for bivariate Appell polynomials II: Non-central limit theorems. Probab. Theory Related Fields 110 333-368.
  • Ho,H. C. and Hsing,T. (1996). On the asymptotic expansion of the empirical process of long memory moving average. Ann. Statist. 24 992-1024.
  • Koul,H. and Surgailis,D. (1997). Asymptotic expansion of M-estimators with long memory errors. Ann. Statist. 25 818-850.
  • Major,P. (1981). Multiple Wiener-It o Integrals. Lecture Notes in Math. 849. Springer, NewYork.
  • Rosenblatt,M. (1961). Independence and dependence. In Proc. Fourth Berkeley Symp. Math. Statist. Probab. 411-443. Univ. California Press, Berkeley.
  • Surgailis,D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22 327-340.
  • Surgailis,D. (1983). On Poisson multiple stochastic integrals and associated equilibrium Markov processes. In Theory and Applications of Random Fields. Lecture Notes in Control and Inform. Sci. 49 233-248. Springer, NewYork.
  • Taqqu,M. S. (1978). A representation for self-similar processes. Stochastic Process. Appl. 7 55-64.
  • Taqqu,M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank.Wahrsch. Verw. Gebiete 50 53-83.