On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields
ESAIM: Probability and Statistics, Tome 14 (2010), pp. 210-255.

Many statistical applications require establishing central limit theorems for sums/integrals ${S}_{T}\left(h\right)={\int }_{t\in {I}_{T}}h\left({X}_{t}\right)\mathrm{d}t$ or for quadratic forms ${Q}_{T}\left(h\right)={\int }_{t,s\in {I}_{T}}\stackrel{^}{b}\left(t-s\right)h\left({X}_{t},{X}_{s}\right)\mathrm{d}s\mathrm{d}t$, where Xt is a stationary process. A particularly important case is that of Appell polynomials h(Xt) = Pm(Xt), h(Xt,Xs) = Pm,n (Xt,Xs), since the “Appell expansion rank” determines typically the type of central limit theorem satisfied by the functionals ST(h), QT(h). We review and extend here to multidimensional indices, along lines conjectured in [F. Avram and M.S. Taqqu, Lect. Notes Statist. 187 (2006) 259-286], a functional analysis approach to this problem proposed by [Avram and Brown, Proc. Amer. Math. Soc. 107 (1989) 687-695] based on the method of cumulants and on integrability assumptions in the spectral domain; several applications are presented as well.

DOI : https://doi.org/10.1051/ps:2008031
Classification : 60F05,  62M10,  60G15,  62M15,  60G10,  60G60
Mots clés : quadratic forms, Appell polynomials, Hölder-Young inequality, Szegö type limit theorem, asymptotic normality, minimum contrast estimation
@article{PS_2010__14__210_0,
author = {Avram, Florin and Leonenko, Nikolai and Sakhno, Ludmila},
title = {On a {Szeg\"o} type limit theorem, the {H\"older-Young-Brascamp-Lieb} inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields},
journal = {ESAIM: Probability and Statistics},
pages = {210--255},
publisher = {EDP-Sciences},
volume = {14},
year = {2010},
doi = {10.1051/ps:2008031},
mrnumber = {2741966},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2008031/}
}
TY  - JOUR
AU  - Avram, Florin
AU  - Leonenko, Nikolai
AU  - Sakhno, Ludmila
TI  - On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields
JO  - ESAIM: Probability and Statistics
PY  - 2010
DA  - 2010///
SP  - 210
EP  - 255
VL  - 14
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2008031/
UR  - https://www.ams.org/mathscinet-getitem?mr=2741966
UR  - https://doi.org/10.1051/ps:2008031
DO  - 10.1051/ps:2008031
LA  - en
ID  - PS_2010__14__210_0
ER  - 
Avram, Florin; Leonenko, Nikolai; Sakhno, Ludmila. On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 210-255. doi : 10.1051/ps:2008031. http://www.numdam.org/articles/10.1051/ps:2008031/

[1] B. Anderson, J.M. Ash, R. Jones, D.G. Rider and B. Saffari, Exponential sums with coefficients 0 or 1 and concentrated Lp norms. Ann. Inst. Fourier 57 (2007) 1377-1404. | Numdam | Zbl 1133.42004

[2] V.V. Anh and N.N. Leonenko, Spectral analysis of fractional kinetic equations with random data. J. Statist. Phys. 104 (2001) 1349-1387. | Zbl 1034.82044

[3] V.V. Anh and N.N. Leonenko, Renormalization and homogenization of fractional diffusion equations with random data. Probab. Theory Relat. Fields 124 (2002) 381-408. | Zbl 1031.60043

[4] V.V. Anh, J.M. Angulo and M.D. Ruiz-Medina, Possible long-range dependence in fractional random fields. J. Statist. Plann. Infer. 80 (1999) 95-110. | Zbl 1039.62090

[5] V.V. Anh, C.C. Heyde and N.N. Leonenko, Dynamic models of long-memory processes driven by Lévy noise. J. Appl. Probab. 39 (2002) 730-747. | Zbl 1016.60039

[6] V.V. Anh, N.N. Leonenko and R. Mcvinish, Models for fractional Riesz-Bessel motion and related processes. Fractals 9 (2001) 329-346.

[7] V.V. Anh, N.N. Leonenko and L.M. Sakhno, Higher-order spectral densities of fractional random fields. J. Stat. Phys. 111 (2003) 789-814. | Zbl 1019.60046

[8] V.V. Anh, N.N. Leonenko and L.M. Sakhno, Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence. Stochastic Methods and their Applications. J. Appl. Probab. A 41 (2004) 35-53. | Zbl 1049.62107

[9] V.V. Anh, N.N. Leonenko and L.M. Sakhno, On a class of minimum contrast estimators. J. Statist. Plann. Infer. 123 (2004) 161-185. | Zbl 1103.62092

[10] F. Avram, On Bilinear Forms in Gaussian Random Variables and Toeplitz Matrices. Probab. Theory Relat. Fields 79 (1988) 37-45. | Zbl 0648.60043

[11] F. Avram, Generalized Szegö Theorems and asymptotics of cumulants by graphical methods. Trans. Amer. Math. Soc. 330 (1992) 637-649. | Zbl 0752.60019

[12] F. Avram and L. Brown, A Generalized Hölder Inequality and a Generalized Szegö Theorem. Proc. Amer. Math. Soc. 107 (1989) 687-695. | Zbl 0696.60035

[13] F. Avram and R. Fox, Central limit theorems for sums of Wick products of stationary sequences. Trans. Amer. Math. Soc. 330 (1992) 651-663. | Zbl 0752.60020

[14] F. Avram and M.S. Taqqu, Noncentral limit theorems and Appell polynomials. Ann. Probab. 15 (1987) 767-775. | Zbl 0624.60049

[15] F. Avram and M.S. Taqqu, Hölder's Inequality for Functions of Linearly Dependent Arguments. SIAM J. Math. Anal. 20 (1989) 1484-1489. | Zbl 0777.26017

[16] F. Avram and M.S. Taqqu, On a Szegö type limit theorem and the asymptotic theory of random sums, integrals and quadratic forms. Dependence in probability and statistics, Lect. Notes Statist. 187. Springer, New York (2006) 259-286. | Zbl 1113.60024

[17] K. Ball, Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. 44 (1991) 351-359. | Zbl 0694.46010

[18] F. Barthe, On a reverse form of the Brascamp-Lieb inequality. Inventiones Mathematicae 134 (2005) 335-361. | Zbl 0901.26010

[19] J. Bennett, A. Carbery, M. Christ and T. Tao, The Brascamp-Lieb inequalities: finiteness, structure and extremals, Geom. Funct. Anal. 17 (2008) 1343-1415. | Zbl 1132.26006

[20] R. Bentkus, On the error of the estimate of the spectral function of a stationary process. Lietuvos Matematikos Rinkinys 12 (1972) 55-71 (In Russian). | Zbl 0253.62050

[21] R. Bentkus, and R. Rutkauskas, On the asymptotics of the first two moments of second order spectral estimators. Liet. Mat. Rink. 13 (1973) 29-45. | Zbl 0283.62088

[22] J. Beran, Statistics for Long-Memory Processes. Chapman & Hall, New York (1994). | Zbl 0869.60045

[23] H.J. Brascamp and E. Lieb, Best constants in Young's inequality, its converse, and its generalization to more than three functions. Adv. Math. 20 (1976) 151-173. | Zbl 0339.26020

[24] P. Breuer and P. Major, Central limit theorems for nonlinear functionals of Gaussian fields. J. Multiv. Anal. 13 (1983) 425-441. | Zbl 0518.60023

[25] P.J. Brockwell, Representations of continuous-time ARMA processes. Stochastic Methods and their Applications. J. Appl. Probab. A 41 (2004) 375-382. | Zbl 1052.60024

[26] E.A. Carlen, E.H. Lieb and M. Loss, A sharp analog of Young's inequality on Sn and related entropy inequalities. J. Geom. Anal. 14 (2004) 487-520. | Zbl 1056.43002

[27] R.L. Dobrushin and P. Major, Non-central limit theorems for non-linear functions of Gaussian fields. Z. Wahrscheinlichkeitstheorie Verw. Geb. 50 (1979) 27-52. | Zbl 0397.60034

[28] R. Fox and M.S. Taqqu, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 (1986) 517-532. | Zbl 0606.62096

[29] R. Fox and M.S. Taqqu, Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Theory Relat. Fields 74 (1987) 213-240. | Zbl 0586.60019

[30] E. Friedgut, Hypergraphs, entropy, and inequalities. Amer. Math. Monthly 111 (2004) 749-760. | Zbl 1187.94017

[31] J. Gao, V.V. Anh and C.C. Heyde, Statistical estimation of nonstationary Gaussian process with long-range dependence and intermittency. Stoch. Process. Appl. 99 (2002) 295-321. | Zbl 1059.60024

[32] R. Gay and C.C. Heyde, On a class of random field models which allows long range dependence. Biometrika 77 (1990) 401-403. | Zbl 0711.62086

[33] M.S. Ginovian, On Toeplitz type quadratic functionals of stationary Gaussian processes. Probab. Theory Relat. Fields 100 (1994) 395-406. | Zbl 0817.60018

[34] M.S. Ginovian and A.A. Sahakyan, Limit theorems for Toeplitz quadratic functionals of continuous-time stationary processes. Probab. Theory Relat. Fields 138 (2007) 551-579. | Zbl 1113.60027

[35] L. Giraitis, Central limit theorem for functionals of linear processes. Lithuanian Math. J. 25 (1985) 43-57. | Zbl 0568.60020

[36] L. Giraitis and D. Surgailis, Multivariate Appell polynomials and the central limit theorem, in Dependence in Probability and Statistics. Edited by E. Eberlein and M.S. Taqqu. Birkhäuser, New York (1986) 21-71. | Zbl 0605.60031

[37] L. Giraitis and D. Surgailis, A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittle estimate. Probab. Theory Relat. Fields 86 (1990) 87-104. | Zbl 0717.62015

[38] L. Giraitis and M.S. Taqqu, Limit theorems for bivariate Appell polynomials, Part 1: Central limit theorems. Probab. Theory Relat. Fields 107 (1997) 359-381. | Zbl 0873.60007

[39] L. Giraitis and M.S. Taqqu, Whittle estimator for finite variance non-Gaussian time series with long memory. Ann. Statist. 27 (1999) 178-203. | Zbl 0945.62085

[40] C.W. Granger and R. Joyeux, An introduction to long-memory time series models and fractional differencing. J. Time Ser. Anal. 10 (1990) 233- 257. | Zbl 0503.62079

[41] V. Grenander and G. Szegö, Toeplitz forms and their applications. University of California Press, Berkeley (1958). | Zbl 0080.09501

[42] C.C. Heyde, Quasi-Likelihood And Its Applications: A General Approach to Optimal Parameter Estimation. Springer-Verlag, New York (1997). | Zbl 0879.62076

[43] C. Heyde and R. Gay, On asymptotic quasi-likelihood. Stoch. Process. Appl. 31 (1989) 223-236. | Zbl 0684.62067

[44] C. Heyde and R. Gay, Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence. Stoch.c Process. Appl. 45 (1993) 169-182. | Zbl 0771.60021

[45] J.R.M. Hosking, Fractional differencing. Biometrika 68 (1981) 165-176. | Zbl 0464.62088

[46] H.E. Hurst, Long-term storage capacity of reservoirs. Trans. Amer. Soc. Civil Eng. 116 (1951) 770-808.

[47] I.A. Ibragimov, On estimation of the spectral function of a stationary Gaussian process. Theory Probab. Appl. 8 (1963) 391-430. | Zbl 0137.12901

[48] I.A. Ibragimov, On maximum likelihood estimation of parameters of the spectral density of stationary time series. Theory Probab. Appl. 12 (1967) 115-119. | Zbl 0173.20703

[49] A.V. Ivanov and N.N. Leonenko, Statistical Analysis of Random Processes. Kluwer Academic Publisher, Dordrecht (1989).

[50] M. Kelbert, N.N. Leonenko and M.D. Ruiz-Medina, Fractional random fields associated with stochastic fractional heat equation. Adv. Appl. Probab. 37 (2005) 108-133. | Zbl 1102.60049

[51] S. Kwapien and W.A. Woyczynski, Random Series and Stochastic Integrals: Single and Multiple. Birkhaäser, Boston (1992). | Zbl 0751.60035

[52] N.N. Leonenko and L.M. Sakhno, On the Whittle estimators for some classes of continuous parameter random processes and fields. Stat. Probab. Lett. 76 (2006) 781-795. | Zbl 1089.62101

[53] E.H. Lieb, Gaussian kernels have only Gaussian maximizers. Invent. Math. 102 (1990) 179-208. | Zbl 0726.42005

[54] V.A. Malyshev, Cluster expansions in lattice models of statistical physics and the quantum theory of fields. Russ. Math. Surveys 35 (1980) 1-62.

[55] I. Niven, Formal power series. Amer. Math. Monthly 76 (1969) 871-889. | Zbl 0184.29603

[56] D. Nualart and G. Peccati, Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 (2005) 177-193. | Zbl 1097.60007

[57] J.G. Oxley, Matroid Theory. Oxford University Press, New York (1992). | Zbl 1115.05001

[58] G. Peccati and C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals. Séminaire de Probabilités XXXVIII. Lect. Notes Math. 1857 247-262. Springer-Verlag, Berlin (2004). | Zbl 1063.60027

[59] H. Reiter and J.D. Stegeman, Classical Harmonic Analysis and Locally Compact Groups. Oxford University Press, USA (2000). | Zbl 0965.43001

[60] W. Rudin, Real and Comlex Analysis. McGraw-Hill, London, New York (1970). | Zbl 0925.00005

[61] W. Rudin, Functional Analysis. McGraw-Hill, London, New York (1991). | Zbl 0867.46001

[62] G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes. Chapman and Hall, New York (1994). | Zbl 0925.60027

[63] V. Solev and L. Gerville-Reache, A sufficient condition for asymptotic normality of the normalized quadratic form Ψn(f,g). C. R. Acad. Sci. Paris, Ser. I 342 (2006) 971-975. | Zbl 1094.62018

[64] R. Stanley, Enumerative combinatorics. Cambridge University Press (1997). | Zbl 0889.05001

[65] E.M. Stein, Singular Integrals and Differential Properties of Functions. Princeton University Press (1970). | Zbl 0207.13501

[66] B. Sturmfels, Grobner bases and convex polytopes. Volume 8 of University lecture Series. AMS, Providence, RI (1996). | Zbl 0856.13020

[67] D. Surgailis, On Poisson multiple stochastic integral and associated Markov semigroups. Probab. Math. Statist. 3 (1984) 217-239. | Zbl 0548.60058

[68] D. Surgailis, Long-range dependence and Appel rank, Ann. Probab. 28 (2000) 478-497. | Zbl 1130.60306

[69] M.S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheorie Verw. Geb. 50 (1979) 53-83. | Zbl 0397.60028

[70] W.T. Tutte, Matroids and graphs. Trans. Amer. Math. Soc. 90 (1959) 527-552. | Zbl 0084.39504

[71] D. Welsh, Matroid Theory. Academic Press, London (1976). | Zbl 0343.05002

[72] W. Willinger, M.S. Taqqu and V. Teverovsky, Stock market prices and long-range dependence. Finance and Stochastics 3 (1999) 1-13. | Zbl 0924.90029

[73] P. Whittle, Hypothesis Testing in Time Series. Hafner, New York (1951).

[74] P. Whittle, Estimation and information in stationary time series. Ark. Mat. 2 (1953) 423-434. | Zbl 0053.41003

[75] A. Zygmund, Trigonometric Series. Volumes I and II. Third edition. Cambridge University Press (2002). | Zbl 1084.42003

Cité par Sources :