Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1973-0315774-6
MathSciNet review: 0315774
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] J. L. Doob, Stochastic processes, Wiley, New York; Chapman & Hall, London, 1953. MR 15,445. MR 0058896 (15:445b)
  • [2] Paul Lévy, Processus stochastiques et mouvement brownien, 2ième éd., Gauthier-Villars, Paris, 1965. MR 32 #8363. MR 0190953 (32:8363)
  • [3] P. Mesani, Orthogonally scattered measures, Advances in Math., vol. 2, Princeton Univ. Press, Princeton, N.J., 1968, pp. 61-117. MR 0228651 (37:4231)
  • [4] L. C. Young, Some new stochastic integrals. I. Analogues of Hardy-Littlewood classes, Advances in Probability 2 (1970), 163-240.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60G05

Retrieve articles in all journals with MSC: 60G05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0315774-6
Keywords: Covariance functions, bounded quadratic variation, decomposition of stochastic process
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society