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Linearly additive random fields with independent increments on time like curves

Takenaka S.

Probability and Mathematical Statistics

2003

Vol. 23, Fasc. 1

1--5

Let V be a convex cone in Rn. A curve L = {l(t); t ∈ R+} ⊂ Rn is called a time-like curve if {l(s); s ≥ t} ⊂ l(t) + V holds for any t. A random field {X(t); t ∈ Rn} whose restriction X|L(t) = X (l(t)) on time-like curve L becomes an additive process is considered and it is characterized as a set-indexed random field on the dual cone V∗.

Some remarks on S alpha S, beta - substable random vectors

Misiewicz J.K., Takenaka S.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 2

253--258

An S α S random vector Xis β-substable, α < β ≤ 2, if Xd = Y Θ 1/β for some symmetric β-stable random vector Y Θ ≥ 0 a random variable with the Laplace transform exp{−tα/β}, Y and Θ are independent. We say that an S α S random vector is maximal if it is not β-substable for any β > α. In the paper we show that the ca,nonical spectral measure for every S α S, β-substablerandom vector X, β > α is equivalent to the Lebesgue measure on Sn−1.We show also that every such vector admits the representation X=Y+Z, where Y is an S α S sub-Gaussian random vector, Z is a maximal S α S random vector, Y and Z are independent. The last representation is not unique.