Book Review: Gaussian processes, function theory, and the inverse spectral problem
Donald E. Sarason · Bulletin of the American Mathematical Society · 1978
A stationary Gaussian process is a continuous map / -> & from the real line into the real L 2 space of a probability measure, P, with the following properties: (i)/£dP = 0forall/; (ii) ƒ£,£, dP depends only on the difference t -s (and so can be written as Q(t -s) 9 where Q is a continuous positive definite function on the line, known as the covariance function of the process); (iii) every function in the linear span of the functions £, is normally distributed.By Bochner's theorem, the covariance function Q admits a representation Q(t)=fe itx dA(x), where A is a positive measure on the line, symmetric with respect to the origin.This leads to what is called the spectral representation of the process: the map sending £, to the function e itx on the line extends to an isometry sending the span, in complex L\P), of the functions £ onto the space Z = L 2 (A).The Gaussian condition (that is, condition (iii)) enables one to give geometric interpretations to various probabilistic aspects of the process.The simplest instance is the statement that, in the L 2 span of the functions £" orthogonality is equivalent to stochastic independence.Because of the spectral representation, one can go a step further, translating probabilistic questions about the process into questions in analysis.The questions in analysis that arise usually involve the theory of Hardy spaces in the upper half-plane and the theory of entire functions of exponential type.It is to them that the book under review is devoted.The process is called deterministic if its past determines its future.This means, in probabilistic terms, that every function £ is measurable with respect to the a-algebra generated by the family {£: s < 0}.Because the process is Gaussian, the latter reduces to the condition that every £, belong to the span, in L 2 (P), of the family {£,: s < 0}.Application of the spectral representation now shows that the process is deterministic if and only if Z is spanned by the functions e isx , s < 0. A criterion is provided by a theorem