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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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