Geodesic
On the Regularity of Spectral Densities
Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 3, pp. 337-340

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Let $W(\theta)$ be an operator function representing the spectral density of a multidimensional stationary random sequence. In the case of finite-dimensional random sequences, it is well known that if $W$ satisfies the Szegö condition $$\int{\log W(\theta)d\theta\geq-cI,}$$ where $c$ is a constant and $I$ the unit operator, then the error of the best linear prediction of a sequence one step ahead will really be nonzero. In the present note, an example is constructed which shows that this assertion is no longer true in the infinite-dimensional case. 
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     author = {Peter D. Lax},
     title = {On the {Regularity} of {Spectral} {Densities}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     year = {1963},
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Peter D. Lax. On the Regularity of Spectral Densities. Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 3, pp. 337-340. http://geodesic.mathdoc.fr/item/TVP_1963_8_3_a11/