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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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On embedding set functions into covariance functions
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by G. D. Allen PDF
Trans. Amer. Math. Soc. 179 (1973), 23-33 Request permission

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.
References
  • J. L. Doob, Stochastic processes, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1953. MR 0058896
  • Paul Lévy, Processus stochastiques et mouvement brownien, Gauthier-Villars & Cie, Paris, 1965 (French). Suivi d’une note de M. Loève; Deuxième édition revue et augmentée. MR 0190953
  • P. Masani, Orthogonally scattered measures, Advances in Math. 2 (1968), 61–117. MR 228651, DOI 10.1016/0001-8708(68)90018-2
  • L. C. Young, Some new stochastic integrals. I. Analogues of Hardy-Littlewood classes, Advances in Probability 2 (1970), 163-240.
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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 179 (1973), 23-33
  • MSC: Primary 60G05
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0315774-6
  • MathSciNet review: 0315774