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Lévy’s Brownian motion; Total positivity structure of M(t)-process and deterministic character

Published online by Cambridge University Press:  22 January 2016

Akio Noda
Akio Noda*
Affiliation:
Department of Mathematics, Aichi University of Education, Kariya 448, Japan
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Let X = {X(A); A ∈ Q} be a Lévy’s Brownian motion with the basic time parameter space Q, where Q is taken to be the n-dimensional metric space Qn,k of constant curvature (2 ≤ n ≤ ∞, — ∞ < k: < ∞), i.e., Q is one of

(a) Euclidean space for k = 0, (b) sphere for k > 0 and

(c) real hyperbolic space for K < 0.

The increment X(A) — X(B) is, by definition, Gaussian in distribution and has mean 0 and variance d(A, B), the distance between A and B. The existence of such a Gaussian random field is well known ([3], [4], [16] and [23]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 94 , June 1984 , pp. 137 - 164
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

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