Advances in Applied Probability

Projective stochastic equations and nonlinear long memory

Ieva Grublytė and Donatas Surgailis

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A projective moving average {Xt, tZ} is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel Q and a linear combination of projections of Xt on `intermediate' lagged innovation subspaces with given coefficients αi and βi,j. The class of such equations includes usual moving average processes and the Volterra series of the LARCH model. Solvability of projective equations is obtained using a recursive equality for projections of the solution Xt. We show that, under certain conditions on Q, αi, and βi,j, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

Article information

Adv. in Appl. Probab., Volume 46, Number 4 (2014), 1084-1105.

First available in Project Euclid: 12 December 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G10: Stationary processes
Secondary: 60F17: Functional limit theorems; invariance principles 60H25: Random operators and equations [See also 47B80]

Projective stochastic equation long memory LARCH model Bernoulli shift invariance principle


Grublytė, Ieva; Surgailis, Donatas. Projective stochastic equations and nonlinear long memory. Adv. in Appl. Probab. 46 (2014), no. 4, 1084--1105. doi:10.1239/aap/1418396244.

Export citation


  • Baillie, R. T. and Kapetanios, G. (2008). Nonlinear models for strongly dependent processes with financial applications. J. Econometrics 147, 60–71.
  • Beran, J. (1994). Statistics for Long-Memory Processes (Monogr. Statist. Appl. Prob. 61). Chapman and Hall, New York.
  • Berkes, I. and Horváth, L. (2003). Asymptotic results for long memory LARCH sequences. Ann. Appl. Prob. 13, 641–668.
  • Davydov, Yu. A. (1970). The invariance principle for stationary process. Theory Prob. Appl. 15, 487–498.
  • Dedecker, J. and Merlevède, F. (2003). The conditional central limit theorem in Hilbert spaces. Stoch. Process. Appl. 108, 229–262.
  • Dedecker, J. \et (2007). Weak Dependence (Lecture Notes Statist. 190). Springer, New York.
  • Doukhan, P., Lang, G. and Surgailis, D. (2012). A class of Bernoulli shifts with long memory: asymptotics of the partial sums process. Preprint. University of Cergy-Pontoise.
  • Doukhan, P., Oppenheim, G. and Taqqu, M. (eds) (2003). Theory and Applications of Long-Range Dependence. Birkhäuser, Boston, MA.
  • Giraitis, L. and Surgailis, D. (2002). ARCH-type bilinear models with double long memory. Stoch. Process. Appl. 100, 275–300.
  • Giraitis, L., Koul, H. L. and Surgailis, D. (2012). Large Sample Inference for Long Memory Processes. Imperial College Press, London.
  • Giraitis, L., Leipus, R. and Surgailis, D. (2009). ARCH($\infty$) models and long memory properties. In Handbook of Financial Time Series, eds T. Mikosch et al., Springer, Berlin, pp. 71–84.
  • Giraitis, L., Robinson, P. M. and Surgailis, D. (2000). A model for long memory conditional heteroskedasticity. Ann. Appl. Prob. 10, 1002–1024.
  • Giraitis, L., Leipus, R., Robinson, P. M. and Surgailis, D. (2004). LARCH, leverage, and long memory. J. Financial Econometrics 2, 177–210.
  • Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Application. Academic Press, New York.
  • Hitczenko, P. (1990). Best constants in martingale version of Rosenthal's inequality. Ann. Prob. 18, 1656–1668.
  • Ho, H.-C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Prob. 25, 1636–1669.
  • Philippe, A., Surgailis, D. and Viano, M.-C. (2006). Invariance principle for a class of non stationary processes with long memory. C. R. Math. Acad. Sci. Paris 342, 269–274.
  • Philippe, A., Surgailis, D. and Viano, M.-C. (2008). Time-varying fractionally integrated processes with nonstationary long memory. Theory Prob. Appl. 52, 651–673.
  • Robinson, P. M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econometrics 47, 67–84.
  • Robinson, P. M. (2001). The memory of stochastic volatility models. J. Econometrics 101, 195–218.
  • Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York.
  • Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitsth. 50, 53–83.
  • Wu, W. B. (2005). Nonlinear system theory: another look at dependence. Proc. Nat. Acad. Sci. USA 102, 14150–14154.
  • Wu, W. B. and Min, W. (2005). On linear processes with dependent innovations. Stoch. Process. Appl. 115, 939–958.
  • Wu, W. B. and Shao, X. (2006). Invariance principles for fractionally integrated nonlinear processes. In Recent Developments in Nonparametric Inference and Probability (IMS Lecture Notes Monogr. Ser. 50), Institute of Mathematical Statistics, Beachwood, OH, pp. 20–30.