Limit theorems for quadratic forms with applications to Whittle’s estimate. (English) Zbl 0940.60037

This paper corrects the limit distribution of a quadratic form statistic previously announced by J. Beran and N. Terrin [Biometrika 83, No. 3, 627-638 (1996; Zbl 0866.62055)]. This difference of quadratic forms can be used in determining the presence of a change-point in the long-memory parameter of a dependent sequence of random variables. The underlying data, \(X_i\), are assumed to be a moving average of iid variables. Define \[ Q^*(x,y) =\sum_{[x]<i,j\leq [y]}b(i-j)\{X_iX_j-r(i-j)\} \] for a suitable sequence of constants \(b(k)\) and where \(r(k)\) denotes the covariance of \(X_0\) and \(X_k\). Define \(Q(x)=Q^*(x,\infty)\) and \[ Z_n(t)=n^{-1/2}t(1-t)\{Q(nt)/nt - Q^*(nt,n)/n(1-t)\}. \] Under decay conditions on the moving average weights and the constants \(b\), the paper establishes a strong approximation of \(Q(x)\) by a Brownian motion, and of \(Z_n\) by a sequence of independent Brownian bridges. It then obtains a strong law of large numbers for \(Q(n)\). The paper examines the case where the moving average coefficients for \(X_k\) depend on a finite-dimensional parameter, \(\lambda_k\). These are assumed to be constant up to a certain point, and then equal to a different constant after that change-point. Splitting the data at \(k\), the Whittle estimate of \(\lambda\) before \(k\) is \(\widehat{\lambda}_k\): that after, \(\widetilde{\lambda}_k\). Theorem 2.1 obtains that under the null hypothesis that there is no change-point, the normalized difference \(\widehat{\lambda}_{nt}-\widetilde{\lambda}_{nt}\) converges weakly to a scaled Brownian motion. Theorem 2.2 provides a similar result for weighted metrics. Theorem 2.3 provides upper tail critical points. The proofs decompose the quadratic forms into martingales and smaller terms.


60F17 Functional limit theorems; invariance principles
60G70 Extreme value theory; extremal stochastic processes
60F05 Central limit and other weak theorems
60F15 Strong limit theorems


Zbl 0866.62055
Full Text: DOI


[1] Anderson, T. W. and Walker, A. M. (1964). On the asy mptotic distribution of the autocorrelations of a sample from a linear process. Ann. Math. Statist. 35 1296-1303. · Zbl 0125.09001
[2] Avram, F. (1988). On bilinear forms in Gaussian random variables and Toeplitz matrices. Probab. Theory Related Fields 79 37-45. · Zbl 0648.60043
[3] Beran, J. and Terrin, N. (1996). Testing for a change of the long-memory parameter. Biometrika 83 627-638. JSTOR: · Zbl 0866.62055
[4] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[5] Cs örg o, M. and Horváth, L. (1993). Weighted Approximations in Probability and Statistics. Wiley, Chichester. · Zbl 0770.60038
[6] Cs örg o, M. and Horváth, L. (1997). Limit Theorems in Change-Point Analy sis. Wiley, Chichester. · Zbl 0884.62023
[7] Cs örg o, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York. · Zbl 0539.60029
[8] de la Pe na, V. and Klass, M. (1994). Order-of-magnitude bounds for expectations involving quadratic forms. Ann. Probab. 22 1044-1077. · Zbl 0810.60012
[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. · Zbl 0586.60019
[10] Giraitis, L. and Surgailis, D. (1990). A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asy mptotic normality of Whittle’s estimate. Probab. Theory Related Fields 86 87-104. · Zbl 0717.62015
[11] Hall, P. and Hey de, C. C. (1980). Martingale Limit Theory and Its Application. Academic Press, New York. · Zbl 0462.60045
[12] Hannan, E. J. (1973). The asy mptotic theory of linear time series models. J. Appl. Probab. 10 130-145. JSTOR: · Zbl 0261.62073
[13] Hannan, E. J. and Hey de, C. C. (1972). On limit theorems for quadratic functions of discrete time series. Ann. Math. Statist. 43 2058-2066. · Zbl 0254.62057
[14] Kouritzin, M. A. (1995). Strong approximation for cross-covariances of linear variables with long-range dependence. Stochastic Process. Appl. 60 343-353. · Zbl 0841.60021
[15] Stout, W. F. (1974). Almost Sure Convergence. Academic Press, New York. · Zbl 0321.60022
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