Random functions with given time correlation

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Abstract

First, on any sequence of real numbers (xλ), λ ϵ [− Λ1 + Λ] ⊂ Z, the pseudo probability Pr(x, x′) of the event xλ ϵ[x, x′[ is defined to be the limit when Λ → ∞ of the ratio of the number of xλ ϵ[x, x′[ to the total number of xλ. The a.d.f. (asymptotic distribution function) of the sequence is then defined by F(x) = Pr(− ∞, x); it possesses the properties of a d.f. (distribution function). Consequently, what is said below applies equally to a sequence of r.v. (random variables) or to a sequence of p.r.v. (pseudorandom variables) consisting of a sequence (nxλ), n ϵ [− N, + N] ⊂ Z of sequences nxλ, λ ϵ [− Λ, + Λ] ⊂ Z.

A weyl's polynomial ϑn(λ) is a polynomial such that one of its coefficieints other than ϑ(0) is irrational. Then any sequence, the fractional part of ϑn(λ), λ ϵ [− Λ, + Λ] ⊂ Z, is asymptotically equidistributed on [0, 1].

A property is given which permits the construction of a sequence (nxλ), n ϵ [− N, + N] ⊂ Z of pseudostochastically independent sequences nxλ, λ ϵ [− Λ, + Λ] ⊂ Z.

It is known that setting Yn = F(− 1)(Xn), it is possible to transform any sequence of r.v. Xn

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