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Linear rescaling of the stochastic process. (English) Zbl 0757.60034
Stochastic processes \(X\) and \(Y\) are considered, whose index has the form \({\mathbf t}=(t_ 1,\dots,t_ k)\), \(t_ i>0\), \(1\leq i\leq k\). It is assumed that the “\({\mathbf t}\)-process” \(\alpha({\mathbf s})Y_{{\mathbf s}\cdot{\mathbf t}}+\beta({\mathbf s})\) has finite-dimensional distributions which converge in law to those of \(X_{{\mathbf t}}\), as \(\min_{1\leq i\leq k} s_ i\) goes to \(\infty\). It is proved that two cases are possible: either \(X\) is essentially a constant process, or the following limits exist \[ \lim_{{\mathbf s}}[\alpha({\mathbf s})\cdot\alpha({\mathbf a}\cdot{\mathbf s})^{- 1}]=A({\mathbf a}),\quad\text{and}\quad\lim_{{\mathbf s}}[\beta({\mathbf s})- \beta({\mathbf a}\cdot{\mathbf s})\cdot\alpha({\mathbf s})\cdot\alpha({\mathbf a}\cdot{\mathbf s})^{-1}]=B({\mathbf a}). \] In the latter case, for all \({\mathbf a}\), the processes \(X_{{\mathbf a}\cdot{\mathbf t}}\) and \(A({\mathbf a})\cdot X_{{\mathbf t}}+B({\mathbf a})\) have the same finite-dimensional distributions. Finally explicit expressions for the functions \(A\) and \(B\) are given, with, typically and respectively, multiplicative and additive forms.
MSC:
60G18 Self-similar stochastic processes
62E20 Asymptotic distribution theory in statistics
60G10 Stationary stochastic processes
60F05 Central limit and other weak theorems
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