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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On embedding set functions into covariance functions

Author: G. D. Allen
Journal: Trans. Amer. Math. Soc. 179 (1973), 23-33
MSC: Primary 60G05
MathSciNet review: 0315774
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Abstract: We consider any continuous hermitian kernel $ M(\Delta ,\Delta ')$ on $ \mathcal{P} \times \mathcal{P}$ where $ \mathcal{P}$ is the prering of intervals of [0,1]. Conditions on M are given to find an interval covariance function $ K(\Delta ,\Delta ')$ so that $ K(\Delta ,\Delta ') = M(\Delta ,\Delta ')$ for all nonoverlapping $ \Delta $ and $ \Delta '$ in $ \mathcal{P}$. The problem is solved by first treating finite hermitian matrices A and finding a positive definite matrix B so that $ {b_{ij}} = {a_{ij}},i \ne j$, so that tr B is minimized. Using natural correspondence between interval covariance functions and stochastic processes, a decomposition theorem is derived for stochastic processes of bounded quadratic variation into an orthogonal process and a process having minimal quadratic variation.

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Keywords: Covariance functions, bounded quadratic variation, decomposition of stochastic process
Article copyright: © Copyright 1973 American Mathematical Society

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