an:01288326
Zbl 0921.60033
Major, P.
Almost sure functional limit theorems. I: The general case
EN
Stud. Sci. Math. Hung. 34, No. 1-3, 273-304 (1998).
00054317
1998
j
60F17 60F15 60G18 28D05
almost sure invariance principle; ergodic theorem; self-similar process; Ornstein-Uhlenbeck process
The aim of this interesting paper is to obtain the almost sure functional limit theorem\break (ASFLT) for arbitrary self-similar process \(X(t,\omega)\). This theorem states that for every bounded measurable functional \(\mathcal F\) on the space \(D[0,1]\) (or \(C[0,1]\) if \(X\) is continuous)
\[
(\log T)^{-1} \int^T_1 \mathcal F(X_t(\cdot ,\omega)) t^{-1} dt\to E \mathcal F(X(\cdot ,\omega))\quad \text{as} T\to \infty
\]
holds for almost all \(\omega\), where \(X_t(u,\omega)=t^{-1/\alpha} X(ut,\omega)\), \(0\leq u\leq 1\), \(t>0\), \(\alpha>0\). The main idea is that ASFLT for \(X(t,\omega)\) follows in a natural way from the ergodic theorem for the generalized Ornstein-Uhlenbeck process corresponding to \(X(t,\omega)\). Next it is shown by an appropriate coupling argument that ASFLT holds also for sequences of random broken lines (or polygons) determined by certain sequences of random variables. Thus the almost sure central limit theorems appearing in the literature of the last decade are special corollaries of the results of this paper.
Tadeusz Inglot (Wroclaw)