Abstract
We obtain variance inequalities for quadratic forms of weakly dependent random variables with bounded fourth moments. We also discuss two applications. Namely, we use these inequalities for deriving the limiting spectral distribution of a random matrix and estimating the long-run variance of a stationary time series.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Andrews, Heteroskedastic and autocorrelation consistent covariancematrix estimation. Econometrica 59, (1991), 817–858.
Z. Bai and J. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd ed. (Springer, New York, 2010).
Z. Bai and W. Zhou, “Large Sample Covariance Matrices without Independence Structures in Columns”, Statist. Sinica 18, (2008), 425–442.
A. Bulinski and A. Shashkin, Limit Theorems for Associated Random Fields and Related Systems (World Scientific, 2007).
X. Chen, “A Note on Moment Inequality for Quadratic Forms”, Statist. and Probab. Lett. 92, 83–88 (2014).
J. Dedecker, P. Doukhan, G. Lang, R. José Rafael León, S. Louhichi, and C. Prieur, Weak Dependence: With Examples and Applications (Springer, New Yor, 2007).
V. F. Gaposhkin, “On the Convergence of Series of Weakly Multiplicative Systems of Functions”, Math. USSR-Sb. 18, 361–371 (1972).
P. Hall and C. C. Heyde, Martingale Limit Theory and Its Application (Academic Press, Boston, 1980).
E. J. Hannan, Multiple Time Series (Wiley, New York, 1970).
B. E. Hansen, “Consistent CovarianceMatrix Estimation for DependentHeterogeneous Processes”, Econometrica 60, 967–972 (1992).
M. Jansson, “Consistent Covariance Matrix Estimation for Linear Processes”, Econometric Theory 18, 1449–1459 (2002).
R. M. Jong and J. Davidson, “Consistency of Kernel Estimators of Heteroscedastic and Autocorrelated Covariance Matrices”, Econometrica 68, (2000), 407–423.
J. Komlós and P. Révész, “Remark to a Paper by Gaposhkin”, Acta Sci. Math. 33, 237–241 (1972).
M. Longecker and R. J. Serfling, “Moment Inequalities for S n under General Dependence Restrictions, with Applications”, Z. Wahrsch. Verw. Gebiete 43, 1–21 (1978).
J. R. Magnus, “The Moments of Products of Quadratic Forms in Normal Variables”, Statistica Neerlandica 32, 201–210 (1978).
F. Moricz, “On the Convergence Properties of Weakly Multiplicative Systems”, Acta Sci. Math. 38, 127–144 (1976).
A. Pajor and L. Pastur, “On the Limiting Empirical Measure of Eigenvalues of the Sum of Rank One Matrices with Log-Concave Distribution”, StudiaMath. 195, 11–29 (2009).
E. Parzen, “On Consistent Estimates of the Spectrum of a Stationary Time Series”, Ann. Math. Statist. 28, 921–932 (1957).
L. Pastur and M. Shcherbina, Eigenvalue distribution of large random matrices, in Mathematical Surveys and Monographs (AmericanMathematical Society, Providence, RI, 2011), Vol. 171.
P. Révész, “M-Mixing Systems”, Acta Sci. Math. 20, 431–442 (1969).
H. White, Asymptotic Theory for Econometricians (Academic Press, Orlando, Fla, 2000).
W. B. Wu and H. Xiao, “Covariance Matrix Estimation for Stationary Time Series”, Ann. Statist. 40, 466–493 (2012).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Yaskov, P. Variance inequalities for quadratic forms with applications. Math. Meth. Stat. 24, 309–319 (2015). https://doi.org/10.3103/S1066530715040055
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530715040055