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Brownian Motion, 2nd Edition by Björn Böttcher, Lothar Partzsch, René L. Schilling

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2 Brownian motion as a Gaussian process

Recall that a one-dimensional random variable Γ is Gaussian if it has the characteristic function

e9783110307290_i0135.jpg

(2.1)

for some real numbers me9783110307290_i0136.jpg and ϭ ≥ 0. If we differentiate (2.1) two times with respect to ξ and set ξ = 0, we see that

e9783110307290_i0137.jpg

(2.2)

A random vector Γ = ( Γ1,..., Γn) ∈e9783110307290_i0138.jpg is Gaussian, if 〈ℓ, Γ〉 is for every ℓ ∈

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