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Some path properties of iterated Brownian motion. (English) Zbl 0789.60060
Cinlar, E. (ed.) et al., Seminar on stochastic processes, 1992. Held at the Univ. of Washington, DC, USA, March 26-28, 1992. Basel: Birkhäuser. Prog. Probab. 33, 67-87 (1992).
The so-called iterated Brownian motion is the process $$Z=(Z(t), t \geq 0)$$ defined by $$Z(t)=X(Y(t))$$, where $$(X(t), t \geq 0)$$, $$(X(-t), t \geq 0)$$ and $$(Y(t), t \geq 0)$$ are three independent standard Brownian motions. The first concern of the paper is to decide whether $$Z$$ specifies $$X$$ and $$Y$$. The answer is surprisingly nearly positive, in the sense that one can construct from $$Z$$ two paths $$(X'(t), -\infty<t<\infty)$$ and $$(Y'(t), t \geq 0)$$ such that $X(t)=X'(t) \text{ for all } t \in (-\infty,\infty) \text{ and } Y(t)=Y'(t) \text{ for all } t \in [0,\infty),$ or $X(t)=X'(-t) \text{ for all } t \in (-\infty,\infty) \text{ and } Y(t)=-Y'(t) \text{ for all } t \in [0,\infty).$ A similar result also holds when $$Z$$ is replaced by $$X \circ X$$. The second concern is an analogue of Khinchin’s law of the iterated logarithm. The author shows that a.s. $\limsup_{t \to 0+} {Z(t) \over t^{1/4} (\log | \log t |)^{3/4}}=2^{5/4}3^{-3/4}.$ (We mention that this last result can also be deduced from the law of the iterated logarithm for a stable subordinator with index 1/4 due to L. Breiman [Ann. Math. Stat. 39, 1818-1824 (1968; Zbl 0192.554)]).
For the entire collection see [Zbl 0780.00020].
Reviewer: J.Bertoin (Paris)

MSC:
 60J65 Brownian motion