A remark on the strong invariance principle for a shot-effect process. (English. Russian original) Zbl 0738.60016

Piterbarg, V. I.

Theory Probab. Math. Stat. 43, 139-141 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 124-126 (1990).

Summary: Let \(\pi_ \lambda(\Delta)\) be a stationary Poisson point process on the line with intensity \(\lambda\), and let \(f(t)\) be a function of bounded variation with finite support. The shot-effect process \(\xi_ \lambda(t)=\int f(t-s)\pi_ \lambda(ds)\) is considered. The strong invariance principle is used to prove a limit theorem of Gnedenko type for the distribution of the variable \(\sup_{[0,T]}\lambda^{- 1/2}(\xi_ \lambda(t)-E\xi_ \lambda(t))\), \(T\to\infty\) and \(\lambda\to\infty\) in such a way that \(\lambda T^{-\alpha}\to\infty\), \(\alpha>0\).

MSC:

60F05	Central limit and other weak theorems
60G10	Stationary stochastic processes

Keywords:

stationary Poisson point process; strong invariance principle; limit theorem