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

Zinchenko, N. M.

Teor. Veroyatn. Mat. Stat., Kiev 38, 51-59 (1988).

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

Keywords:

speed of growth; law of iterated logarithm; stable random field