## The Annals of Probability

### Central limit theorems for quadratic forms with time-domain conditions

#### Abstract

We establish the central limit theorem for quadratic forms $\Sigma_{t, s=1}^N b(t - s)P_{m, n} (X_t, X_s)$ of the bivariate Appell polynomials $P_{m, n} (X_t, X_x))$ under time-domain conditions. These conditions relate the weights $b(t)$ and the covariances of the sequences $(P_{m, n} (X_t, X_s))$ and $(X_t)$. The time-domain approach, together with the spectral domain approach developed earlier, yields a general set of conditions for central limit theorems.

#### Article information

Source
Ann. Probab., Volume 26, Number 1 (1998), 377-398.

Dates
First available in Project Euclid: 31 May 2002

https://projecteuclid.org/euclid.aop/1022855425

Digital Object Identifier
doi:10.1214/aop/1022855425

Mathematical Reviews number (MathSciNet)
MR1617055

Zentralblatt MATH identifier
0943.60018

#### Citation

Giraitis, Liudas; Taqqu, Murad S. Central limit theorems for quadratic forms with time-domain conditions. Ann. Probab. 26 (1998), no. 1, 377--398. doi:10.1214/aop/1022855425. https://projecteuclid.org/euclid.aop/1022855425

#### References

• [1] Arcones, M. A. (1994). Limit theorems for non-linear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242-2274.
• [2] Avram, F. (1988). On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Related Fields 79 37-46.
• [3] Avram, F. (1992). Generalized Szeg¨o theorems and asymptotics of cumulants by graphical methods. Trans. Amer. Math. Soc. 330 637-649.
• [4] Avram, F. and Fox, R. (1992). Central limit theorems for sums of Wick products of stationary sequences. Trans. Amer. Math. Soc. 330 651-663.
• [5] Avram, F. and Taqqu, M. S. (1987). Noncentral limit theorems and Appell polynomials. Ann. Probab. 15 767-775.
• [6] Breuer, P. and Major, P. (1985). Central limit theorems for non-linear functionals of Gaussian fields. J. Multivariate Anal. 13 425-441.
• [7] Fox, R. and Taqqu, M. S. (1985). Non-central limit theorems for quadratic forms in random variables having long-range dependence. Ann. Probab. 13 428-446.
• [8] Fox, R. and Taqqu, M. S. (1986). Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Ann. Statist. 14 517-532.
• [9] Fox, R. and Taqqu, M. S. (1987). Central limit theorems for quadratic forms in random variables having long-range dependence. Probab. Theory Related Fields 74 213-240.
• [10] Giraitis, L. (1985). Central limit theorem for functionals of a linear process. Lithuanian Math. J. 25 25-35.
• [11] Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Probab. Theory Related Fields 70 191-212.
• [12] Giraitis, L. and Surgailis, D. (1986). Multivariate Appell polynomials and the central limit theorem. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 21-71. Birkh¨auser, Boston.
• [13] Giraitis, L. and Taqqu, M. S. (1996). Whittle estimator for non-Gaussian long-memory time series. Preprint.
• [14] Giraitis, L. and Taqqu, M. S. (1997). Limit theorems for bivariate Appell polynomials. I. Central limit theorems. Probab. Theory Related Fields 107 359-381.
• [15] Giraitis, L., Taqqu, M. S. and Terrin, N. (1997). Limit theorems for bivariate Appell polynomials. II. Non-central limit theorems. Probab. Theory Related Fields. To appear.
• [16] Ho, H. C. (1992). On limiting distributions of nonlinear functions of noisy Gaussian sequences. Stochastic Anal. Appl. 10 417-430.
• [17] Ho, H. C. and Sun, T. C. (1987). A central limit theorem for non-instantaneous filters of a stationary Gaussian process. J. Multivariate Anal. 22 144-155.
• [18] Ho, H. C. and Sun, T. C. (1990). Limiting distributions of nonlinear vector functions of stationary Gaussian processes. Ann. Probab. 18 1159-1173.
• [19] Malyshev, V. A. (1980). Cluster expansions in lattice models of statistical physics and the quantum theory of fields. Russian Math. Surveys 35 1-62.
• [20] Shohat, J. (1936). The relation of the classical orthogonal polynomials to the polynomials of Appell. Amer. J. Math. 58 453-464.
• [21] Sun, T. C. (1963). A central limit theorem for non-linear functions of a normal stationary process. Journal of Mathematics and Mechanics 12 945-978.
• [22] Sun, T. C. (1965). Some further results on central limit theorems for non-linear functions of a normal stationary process. Journal of Mathematics and Mechanics 14 71-85.
• [23] Surgailis, D. (1983). On Poisson multiple stochastic integral and associated equilibrium Markov process. Theory and Applications of Random Fields. Lecture Notes in Control and Inform. Sci. 49. Springer, Berlin.