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# Certain Positive-Definite Kernels

Mina Ossiander and Edward C. Waymire
Proceedings of the American Mathematical Society
Vol. 107, No. 2 (Oct., 1989), pp. 487-492
DOI: 10.2307/2047839
Stable URL: http://www.jstor.org/stable/2047839
Page Count: 6
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## Abstract

In one way or another, the extension of the standard Brownian motion process $\{B_t: t \in \lbrack 0, \infty)\}$ to a (Gaussian) random field $\{B_t: \mathbf{t} \in \mathbf{R}^d_+ \}$ involves a proof of the positive semi-definiteness of the kernel used to generalize $\rho(s, t) = \operatorname{cov}(B_s, B_t) = s \wedge t$ to multidimensional time. Simple direct analytical proofs are provided here for the cases of (i) the Levy multiparameter Brownian motion, (ii) the Chentsov Brownian sheet, and (iii) the multiparameter fractional Brownian field.

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