On the question of the asymptotics of the increments of asymmetric stable processes. (English. Russian original) Zbl 0840.60038

Zinchenko, N. M.

Theory Probab. Math. Stat. 46, 51-57 (1993); translation from Teor. Jmovirn. Mat. Stat. 46, 53-60 (1992).

An integral criterion is obtained for answering the question whether \(f(T) = a_T^{1/ \alpha} \varphi (T)\) is an upper or lower function for increments of the stable process \(\xi_{\alpha, \beta} (t)\) with parameters \(\alpha \in (1,2)\) and \(\beta = - 1\) on the intervals of length \(a_T\) \((a_T\) is a nondecreasing function on \([0, \infty)\) such that \(0 < a_T \leq T\) and \(T/a_T\) does not decrease as \(T\) increases). The paper develops the previous results of the author [Theory Probab. Appl. 32, No. 4, 724-727 (1987); translation from Teor. Veroyatn. Primen. 32, No. 4, 793-796 (1987; Zbl 0654.60027)]. Recall that a stable distribution \(G\) has the following exponential estimate \(1 - G(x) \sim C_\alpha x^{- \lambda/2} \exp (-B x^\lambda)\) for \(x\) large enough. It is proved that regular function \(f(t)\) is an upper function for increments if \[ I = \int^\infty_1 \varphi (t)^{- \lambda/2} \exp \bigl(- B \varphi (t)^\lambda \bigr) {dt \over a_t} < \infty, \] and it is a lower function if \(I = \infty\). In studying the question of upper functions, the rejection of assumptions on regularity of \(f(t)\) gives rise to conditions on the rate of increase of \(a_T\) and \(\varphi (T)\) more restrictive than the previous result.

Reviewer: A.I.Volodin (Kazan’)

MSC:

60G17	Sample path properties
60F15	Strong limit theorems

Keywords:

upper and lower function of a process; increments of a process; stable process with parameters

Citations:

Zbl 0715.60041; Zbl 0654.60027