Asymptotic of increments of stable stochastic processes with one-signed jumps. (Russian) Zbl 0654.60027

Zinchenko, N. M.

Teor. Veroyatn. Primen. 32, No. 4, 793-796 (1987).

Let \(\xi\) be a stable stochastic process with negative jumps and \(\alpha\in (1,2)\). The author gives conditions under which \[ \overline{\lim}_{T\to \infty}\sup_{0\leq t\leq T-a_ T}\gamma_ T(\xi (t+a_ T)-\xi (t))=1,\quad \overline{\lim}_{T\to \infty}\gamma_ T(\xi (T+a_ T)-\xi (T))=1, \] with probability 1, for some nondecreasing function \(a_ T\) and some decreasing to zero function \(\gamma_ T\). The same result is obtained for partial sums of i.i.d. r.v.’s from the domain of attraction of a stable law which satisfy the “strong invariance principle”.

Reviewer: K.Kubilius

MSC:

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

Keywords:

maximum inequality; stable stochastic process; strong invariance principle