## On asymptotics of stable random fields.(Russian)Zbl 0665.60055

Let $$\xi$$ (t), $$t=(t_ 1,...,t_ n)$$, be a stable random field with parameters $$\beta =-1$$, $$1<\alpha <2$$, and $$\mu$$ ($$\cdot)$$ the Lebesgue measure in $$R^ n$$. The paper deals with the asymptotic behaviour of $$\xi$$ (t) and its increments $$\xi$$ (Q) over n-dimensional rectangles Q when $$\prod^{n}_{i=1}t_ i$$ or $$\mu$$ (Q)$$\to \infty$$ but not all $$t_ i\to \infty.$$
For instance, let $$b_ T\geq T^{1/n}$$ and $$0<a_ T<T$$ be nondecreasing functions satisfying suitable growth conditions. Denote $D_ T=\{t\in R^ n_+:\quad 0\leq t_ i\leq b_ T,\quad \prod^{n}_{i=1}t_ i\leq T\}$ and let $$L_ T$$ be the class of intervals $$Q\subset D_ T$$ with $$\mu (Q)\leq a_ T$$. Then $\overline{\lim}_{T\to \infty}\sup_{Q\subset L_ T}\gamma_ T\xi (Q)=1\quad a.s.,\quad where$
$\gamma_ T=a_ T^{-1/\alpha}\{B^{-1}_{\alpha}(\ln \ln T+\ln T a_ T^{-1}+\ln (\ln b_ T a_ T^{-1/n}+1)^{n-1}\}^{- 1+1/\alpha}$ and the constant $$B_{\alpha}$$ depends only on $$\alpha$$. The method of proof goes back to one proposed by Csörgö and Révész.
Reviewer: N.M.Zinchenko

### MSC:

 60G60 Random fields 60J99 Markov processes 60G17 Sample path properties 60F15 Strong limit theorems