On the central limit theorem for Toeplitz quadratic forms

A. A. Sahakian   ; M. S. Ginovyan

Teoriâ veroâtnostej i ee primeneniâ, Tome 49 (2004) no. 4, pp. 653-671

Résumé

Let $X(t)$, $t = 0,\pm1,\ldots$, be a real-valued stationary Gaussian sequence with a spectral density function $f(\lambda)$. The paper considers the question of applicability of the central limit theorem (CLT) for a Toeplitz-type quadratic form $Q_n$ in variables $X(t)$, generated by an integrable even function $g(\lambda)$. Assuming that $f(\lambda)$ and $g(\lambda)$ are regularly varying at $\lambda=0$ of orders $\alpha$ and $\beta$, respectively, we prove the CLT for the standard normalized quadratic form $Q_n$ in a critical case $\alpha+\beta=\frac{1}{2}$. We also show that the CLT is not valid under the single condition that the asymptotic variance of $Q_n$ is separated from zero and infinity.

Détail

Keywords: stationary Gaussian sequence, spectral density, Toeplitz-type quadratic forms, central limit theorem, asymptotic variance, slowly varying functions.

Bibliographie

[1] Avram F., “On bilinear forms in Gaussian random variables and Toeplitz matrices”, Probab. Theory Related Fields, 79:1 (1988), 37–45 | DOI | MR | Zbl

[2] Bentkus R., “Ob oshibke otsenki spektralnoi funktsii statsionarnogo protsessa”, Litov. matem. sb., 12:1 (1972), 55–71 | MR | Zbl

[3] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, v. 2, Mir, M., 1984, 752 pp. | MR

[4] Fox R., Taqqu M. S., “Central limit theorem for quadratic forms in random variables having long-range dependence”, Probab. Theory Related Fields, 74:2 (1987), 213–240 | DOI | MR | Zbl

[5] Ginovyan M. S., “Asimptoticheski effektivnoe neparametricheskoe otsenivanie funktsionalov ot spektralnoi plotnosti, imeyuschei nuli”, Teoriya veroyatn. i ee primen., 33:2 (1988), 315–322 | MR | Zbl

[6] Ginovyan M. S., “Zamechanie o tsentralnoi predelnoi teoreme dlya kvadratichnykh form tëplitseva tipa ot statsionarnykh gaussovskikh velichin”, Izv. NAN Armenii, ser. matem., 28:2 (1993), 78–81 | MR | Zbl

[7] Ginovian M. S., “On Toeplitz type quadratic functionals of stationary Gaussian process”, Probab. Theory Related Fields, 100:3 (1994), 395–406 | DOI | MR | Zbl

[8] Giraitis L., Surgailis D., “A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle's estimate”, Probab. Theory Related Fields, 86:1 (1990), 87–104 | DOI | MR | Zbl

[9] Grenander U., Segë G., Tëplitsevy formy i ikh prilozheniya, IL, M., 1961, 308 pp. | MR

[10] Hasminskii R. Z., Ibragimov I. A., “Asymptotically efficient nonparametric estimation of functionals of a spectral density function”, Probab. Theory Related Fields, 73:3 (1986), 447–461 | DOI | MR | Zbl

[11] Ibragimov I. A., “Ob otsenke spektralnoi funktsii statsionarnogo gaussovskogo protsessa”, Teoriya veroyatn. i ee primen., 8:4 (1963), 391–430 | Zbl

[12] Rosenblatt M., “Asymptotic behavior of eigenvalues of Toeplitz forms”, J. Math. Mech., 11:6 (1962), 941–949 | MR | Zbl

[13] Seneta E., Pravilno menyayuschiesya funktsii, Nauka, M., 1976, 141 pp. | MR

[14] Terrin N., Taqqu M. S., “A noncentral limit theorem for quadratic forms of Gaussian stationary sequences”, J. Theoret. Probab., 3:3 (1990), 449–475 | DOI | MR | Zbl

[15] Terrin N., Taqqu M. S., “Convergence in distribution of sums of bivariate Appell polynomials with long-range dependence”, Probab. Theory Related Fields, 90:1 (1991), 57–81 | DOI | MR | Zbl

[16] Terrin N., Taqqu M. S., “Convergence to a Gaussian limit as the normalization exponent tends to $\frac12$”, Statist. Probab. Lett., 11:5 (1991), 419–427 | DOI | MR | Zbl

[17] Taniguchi M., “Berry–Esseen theorems for quadratic forms of Gaussian stationary processes”, Probab. Theory Related Fields, 72:2 (1986), 185–194 | DOI | MR | Zbl

[18] Taniguchi M., Kakizawa Y., Asymptotic Theory of Statistical Inference for Time Series, Springer-Verlag, New York, 2000, 661 pp. | MR