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Global Strassen-type theorems for iterated Brownian motions. (English) Zbl 0843.60072

Let \(W_1(t)\), \(t \in R\), and \(W_2(t)\), \(t \geq 0\), be two independent standard Wiener processes. Then the process \(Z(t) = W_1(W_2(t))\), \(t \geq 0\), is called iterated Brownian motion. The authors generalize this notion to a class of so-called iterated processes and prove for them global \((t \to \infty)\), as well as local \((t \to 0)\), LIL type results. They apply Strassen’s method by proving a joint functional limit theorem for a pair of independent Wiener processes. Several interesting results are derived as corollaries of this general theorem. Among them there is the following global version of K. Burdzy’s theorem [in: Séminaire de probabilités XXVII. Lect. Notes Math. 1557, 177-181 (1993; Zbl 0789.60061) and Lett. Math. Phys. 27, No. 3, 239-241 (1993; Zbl 0773.60078)]: \[ \lim_{T \to \infty} [Z(T)/T^{1/4} (\log \log T)^{3/4}] = 2^{5/4} 3^{-3/4}\quad \text{a.s.} \] Similar results are also proved for iterated random walks via invariance.

MSC:

60J65 Brownian motion
60F15 Strong limit theorems
60F17 Functional limit theorems; invariance principles
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