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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The measurability of a stochastic process of second order and its linear space


Author: Stamatis Cambanis
Journal: Proc. Amer. Math. Soc. 47 (1975), 467-475
DOI: https://doi.org/10.1090/S0002-9939-1975-0356206-9
MathSciNet review: 0356206
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Abstract: It is of considerable theoretical and practical interest to know whether a stochastic process has a measurable modification. For the important class of second order processes, simple necessary and sufficient conditions for the existence of a measurable modification are given in terms of the autocorrelation of the process and the separability of its reproducing kernel Hilbert space or its linear space. It is shown that weakly continuous processes, processes with orthogonal increments and second order martingales always have measurable modifications. Also necessary and sufficient conditions are given in terms of integral representations for the linear space of a second order process to be separable. As a consequence it is shown that a second order process is oscillatory if and only if its linear space is separable.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0356206-9
Keywords: Second order processes, measurable processes, weakly continuous processes, processes with orthogonal increments, second order martingales, the linear space of a process, integral representations, oscillatory processes
Article copyright: © Copyright 1975 American Mathematical Society

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