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Iterated Brownian motion and stable $$(1/4)$$ subordinator. (English) Zbl 0854.60082
Let $$B_t$$, $$B^+_t$$, $$B^-_t$$, $$t \geq 0$$, be three independent linear Brownian motions started from 0. Denote $$S_t = \sup \{B_s :0 \leq s \leq t\}$$ and $$\sigma_t = \inf \{u : S_u > t\}$$. Consider also the supremum processes $$S^+$$, $$S^-$$ of $$B^+$$ and $$B^-$$, respectively, denote $$\sigma^+$$, $$\sigma^-$$ the right-continuous inverses of $$B^+$$ and $$B^-$$, respectively. The author notices that the compound process $$\sigma \circ \sigma^+_t$$ is a stable subordinator with index $$\alpha = 1/4$$ and proposes a very easy method to investigate the path behaviour of iterated Brownian motion $$X_t$$. Here the process $$X_t$$, $$t \geq 0$$, is given by expressions $$X_t = B^+ (B_t)$$ if $$B_t \geq 0$$ and $$X_t = B^-(- B_t)$$ if $$B_t < 0$$. Based on known properties of stable processes with $$\alpha \in (0,1)$$ the LIL for $$X_t$$ is of the form: $\limsup t^{-1/4} (\log |\log t |)^{- 3/4} X_t = 2^{5/4} 3^{-3/4}.$ Both as $$t \to 0$$ and $$t \to \infty$$ are proved. Also the integral tests to study the rate of grows for $$S^+ \circ S$$ and $$X_t$$ are proposed and modulas of continuity of the supremum of iterated Brownian motion are obtained.

MSC:
 60J65 Brownian motion 60J99 Markov processes 60G17 Sample path properties
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References:
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