Functional limit theorem for the shot noise processes. (Ukrainian, English) Zbl 1026.60036

Il’yenko, A. B.

Teor. Jmovirn. Mat. Stat. 65, 46-52 (2001); translation in Theory Probab. Math. Stat. 65, 53-60 (2002).

Let \(\zeta(t),t\in\mathbb R\), be a homogeneous stochastic process with independent increments and let \(g(x)\in L_2(\mathbb R)\). The integral \(\theta(u)=\int_{-\infty}^{\infty}g(u-s)d\zeta(s), u\in\mathbb R\), determines a stationary shot noise process. The author investigates convergence in distribution as \(T\to\infty\) of the normed processes \(\theta_T(u)=\int_0^{tT}\zeta(u)du, t\in[0,1],\) to the fractional Brownian motion in the space \(C[0,1]\).

Reviewer: Mikhail Moklyachuk (Kyïv)

MSC:

60F17	Functional limit theorems; invariance principles
60G10	Stationary stochastic processes

Keywords:

shot noise process; fractional Brownian motion; process with independent increments