×

Trace inequalities for Toeplitz matrices and applications to Gaussian probabilities. (Inégalités de trace pour des matrices de Toeplitz et applications à des vraisemblances gaussiennes.) (French) Zbl 0781.60019

Summary: Let \(u\) be an integrable function on the 1-dimensional torus and \(T_ n(u)\) be the Toeplitz matrix with entries \(\widehat{u}(s-t)\), \(0 \leq s\), \(t\leq n-1\), where \(\widehat{u}\) is the Fourier transform of \(u\). It is shown that if \(u_ 1,\dots,u_ r\) are in the Banach algebra of those \(u\) that satisfy \(\| u\| = \| u\|_ \infty + \| u\|_{1/2} < \infty\), where \(\| u\|_ \infty\) is the \(L^ \infty\)-norm of \(u\) and \(\| u\|_{1/2} = (\sum^{+\infty}_{- \infty}| t| | \widehat{u}(t)|^ 2)^{1/2}\), then \[ \| T_ n(u_ 1\dots u_ r)-T_ n(u_ 1)\dots T_ n(u_ r)\|_ 1 \leq \sum_{i<j}\| u_ i\|_{1/2}\| u_ j\|_{1/2}\prod_{k\neq i,j}\| u_ k\|_ \infty, \] where the norm on the left is the trace class norm. Using the inequality \(|\text{tr}(A)| \leq \| A\|_ 1\) (tr for trace), it is shown that if boundedness is replaced by continuity, then \(\text{tr}(T_ n(u_ 1\dots u_ r) - T_ n(u_ 1)\dots T_ n(u_ r))\) is convergent \((n\to \infty)\). These results are used to study Whittle’s approximation error for log-likelihoods of stationary Gaussian sequences. It is shown that its moments are bounded or convergent under suitable conditions for spectral densities.

MSC:

60E15 Inequalities; stochastic orderings
60G15 Gaussian processes
PDFBibTeX XMLCite