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 ST(h)=∫t∈ITh(Xt)dt or for quadratic forms QT(h)=∫t,s∈ITb^(t-s)h(Xt,Xs)dsdt, 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 : 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 = {https://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
SP  - 210
EP  - 255
VL  - 14
PB  - EDP-Sciences
UR  - https://numdam.org/articles/10.1051/ps:2008031/
DO  - 10.1051/ps:2008031
LA  - en
ID  - PS_2010__14__210_0
ER  - 
%0 Journal Article
%A Avram, Florin
%A Leonenko, Nikolai
%A Sakhno, Ludmila
%T On a Szegö type limit theorem, the Hölder-Young-Brascamp-Lieb inequality, and the asymptotic theory of integrals and quadratic forms of stationary fields
%J ESAIM: Probability and Statistics
%D 2010
%P 210-255
%V 14
%I EDP-Sciences
%U https://numdam.org/articles/10.1051/ps:2008031/
%R 10.1051/ps:2008031
%G en
%F PS_2010__14__210_0
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. https://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

[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

[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

[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

[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

[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

[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

[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

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

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

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

[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

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

[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

[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

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

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

[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

[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

[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

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

[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

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

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

[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

[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

[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

[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

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

[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

[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

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

[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

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

[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

[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

[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

[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

[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

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

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

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

[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

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

[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

[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

[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

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

[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

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

[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

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

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

[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

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

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

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

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

[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

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

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

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

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

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

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

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

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

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

[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

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

Cité par Sources :