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A stochastic process X = {Xt: t ~ T} is called spherically generated if for each random vector X= (Xtl,...,X,n), there exist a random vector Y=(Y1,---, Ym) with a spherical (radially symmetric) distribution and a matrix A such that X is distributed as AY. X is said to have the linear regression property if ~(XoIX 1 .... ,X,) is a linear function of X1,...,X . whenever the Xj's are elements of the linear span of X. It is shown that providing the linear span of X has dimension larger than two, then X has the linear regression property if and only if it is spherically generated. The class of symmetric stable processes which are spherically generated is shown to coincide with the class of socalled sub-Gaussian processes, characterizing those stable processes having the linear regression property.

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