TESTING GAUSSIAN SEQUENCES AND ASYMPTOTIC INVERSION OFTOEPLITZ OPERATORS
Malek Bouaziz
Abstract: This paper is motivated by the statistical problem of testing a zero-mean stationary
Gaussian probability measure on against a similiar probability The method uses
a sequence of Neyman-Pearson’s tests of the finite sections of against the
corresponding sections of First, following D. Dacunha-Castelle, we discuss the
behaviour of the power achieved for levels approaching zero exponentially fast at a suitable
rate. Then, considering the likelihood ratio of w.r.t. , we ask whether there exist
approximate inverses of the covariance matrices of these probabilities, and approximates of
their determinants, which preserve the asymptotics of the tests considered. It turns
out that these matrices are finite sections of the Toeplitz operators whose symbols
are the spectral densities of and Using results of H. Widom on this class
of operators we point out that such approximations exist and work under some
factorisation condition for spectral densities. It is also shown that the same approximation
method works for asymptotic solving of a class of discrete Wiener-Hopf equations.
on 
against a similiar probability 
The method uses
a sequence of Neyman-Pearson’s tests of the finite sections 
of 
against the
corresponding sections 
of 
First, following D. Dacunha-Castelle, we discuss the
behaviour of the power achieved for levels approaching zero exponentially fast at a suitable
rate. Then, considering the likelihood ratio of 
w.r.t. 
, we ask whether there exist
approximate inverses of the covariance matrices of these probabilities, and approximates of
their determinants, which preserve the asymptotics of the tests considered. It turns
out that these matrices are finite sections of the Toeplitz operators whose symbols
are the spectral densities of 
and 
Using results of H. Widom on this class
of operators we point out that such approximations exist and work under some
factorisation condition for spectral densities. It is also shown that the same approximation
method works for asymptotic solving of a class of discrete Wiener-Hopf equations.