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Projective stochastic equations and nonlinear long memory. (English) Zbl 1305.60026
Summary: A projective moving average \(\{X_t: t\in \mathbb{Z}\}\) 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 \(X_t\) on ‘intermediate’ lagged innovation subspaces with given coefficients \(\alpha_i\) and \(\beta_{i,j}\). The class of such equations includes the 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 \(X_t\). We show that, under certain conditions on \(Q\), \(\alpha_i\) and \(\beta_{i,j}\), this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process.

MSC:
60G10 Stationary stochastic processes
60F17 Functional limit theorems; invariance principles
60H25 Random operators and equations (aspects of stochastic analysis)
Software:
longmemo
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[1] Baillie, R. T. and Kapetanios, G. (2008). Nonlinear models for strongly dependent processes with financial applications. J. Econometrics 147 , 60-71. · Zbl 1429.62382
[2] Beran, J. (1994). Statistics for Long-Memory Processes (Monogr. Statist. Appl. Prob. 61 ). Chapman and Hall, New York. · Zbl 0869.60045
[3] Berkes, I. and Horváth, L. (2003). Asymptotic results for long memory LARCH sequences. Ann. Appl. Prob. 13 , 641-668. · Zbl 1032.62078
[4] Davydov, Yu. A. (1970). The invariance principle for stationary process. Theory Prob. Appl. 15 , 487-498. · Zbl 0219.60030
[5] Dedecker, J. and Merlevède, F. (2003). The conditional central limit theorem in Hilbert spaces. Stoch. Process. Appl. 108 , 229-262. · Zbl 1075.60501
[6] Dedecker, J. \et (2007). Weak Dependence (Lecture Notes Statist. 190 ). Springer, New York.
[7] 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.
[8] Doukhan, P., Oppenheim, G. and Taqqu, M. (eds) (2003). Theory and Applications of Long-Range Dependence. Birkhäuser, Boston, MA. · Zbl 1005.00017
[9] Giraitis, L. and Surgailis, D. (2002). ARCH-type bilinear models with double long memory. Stoch. Process. Appl. 100 , 275-300. · Zbl 1057.62070
[10] Giraitis, L., Koul, H. L. and Surgailis, D. (2012). Large Sample Inference for Long Memory Processes. Imperial College Press, London. · Zbl 1279.62016
[11] 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. · Zbl 1178.91160
[12] Giraitis, L., Robinson, P. M. and Surgailis, D. (2000). A model for long memory conditional heteroskedasticity. Ann. Appl. Prob. 10 , 1002-1024. · Zbl 1084.62516
[13] Giraitis, L., Leipus, R., Robinson, P. M. and Surgailis, D. (2004). LARCH, leverage, and long memory. J. Financial Econometrics 2 , 177-210.
[14] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Application. Academic Press, New York. · Zbl 0462.60045
[15] Hitczenko, P. (1990). Best constants in martingale version of Rosenthal’s inequality. Ann. Prob. 18 , 1656-1668. · Zbl 0725.60018
[16] Ho, H.-C. and Hsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Prob. 25 , 1636-1669. · Zbl 0903.60018
[17] 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. · Zbl 1086.60506
[18] Philippe, A., Surgailis, D. and Viano, M.-C. (2008). Time-varying fractionally integrated processes with nonstationary long memory. Theory Prob. Appl. 52 , 651-673. · Zbl 1167.60326
[19] Robinson, P. M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econometrics 47 , 67-84. · Zbl 0734.62070
[20] Robinson, P. M. (2001). The memory of stochastic volatility models. J. Econometrics 101 , 195-218. · Zbl 0966.62079
[21] Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York. · Zbl 0321.60022
[22] Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitsth. 50 , 53-83. · Zbl 0397.60028
[23] Wu, W. B. (2005). Nonlinear system theory: another look at dependence. Proc. Nat. Acad. Sci. USA 102 , 14150-14154. · Zbl 1135.62075
[24] Wu, W. B. and Min, W. (2005). On linear processes with dependent innovations. Stoch. Process. Appl. 115 , 939-958. · Zbl 1081.62071
[25] 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. · Zbl 1268.60045
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