A Few Remarks on the Operator Norm of Random Toeplitz Matrices

We present some results concerning the almost sure behavior of the operator norm of random Toeplitz matrices, including the law of large numbers for the norm, normalized by its expectation (in the i.i.d. case). As tools we present some concentration inequalities for suprema of empirical processes, which are refinements of recent results by Einmahl and Li.

[1]  S. Nagaev,et al.  Probability inequalities for sums of independent random variables with values in a Banach space , 1971 .

[2]  J. Kuelbs,et al.  An inequality for the law of the iterated logarithm , 1983 .

[3]  M. Talagrand,et al.  Probability in Banach Spaces: Isoperimetry and Processes , 1991 .

[4]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[5]  M. Talagrand.  New concentration inequalities in product spaces , 1996 .

[6]  Вероятностные неравенства для сумм независимых случайных величин в терминах урезанных псевдомоментов@@@Probabilistic inequalities for sums of independent random variables in terms of truncated pseudomoments , 1997 .

[7]  Z. Bai.  METHODOLOGIES IN SPECTRAL ANALYSIS OF LARGE DIMENSIONAL RANDOM MATRICES , A REVIEW , 1999 .

[8]  J. Zinn,et al.  Exponential and Moment Inequalities for U-Statistics , 2000, math/0003228.

[9]  A. Dembo,et al.  Spectral measure of large random Hankel, Markov and Toeplitz matrices , 2003, math/0307330.

[10]  Steven J. Miller,et al.  Distribution of Eigenvalues for the Ensemble of Real Symmetric Toeplitz Matrices , 2005 .

[11]  E. Rio,et al.  Concentration around the mean for maxima of empirical processes , 2005, math/0506594.

[12]  U. Einmahl,et al.  Characterization of LIL behavior in Banach space , 2006, math/0608687.

[13]  M. Meckes.  On the spectral norm of a random Toeplitz matrix , 2007, math/0703134.

[14]  Arup Bose,et al.  Spectral norm of random large dimensional noncentral Toeplitz and Hankel matrices , 2007 .

[15]  Journal Url,et al.  A tail inequality for suprema of unbounded empirical processes with applications to Markov chains , 2008 .