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A multi- dimensional extension of the arcsine law. (Une extension multidimensionnelle de la loi de l’arc sinus.) (French) Zbl 0738.60072
Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 294-314 (1989).
[For the entire collection see Zbl 0722.00030.]
Lévy’s arcsine law of the occupation time of the positive demi-line of the BM has beautiful extensions in this paper. The underlying model is the Walsh process [J. B. Walsh, Astérisque 52-53, 37-45 (1978; Zbl 0385.60063)], i.e. Bessel process of order $$\mu$$ on $$k$$ demi-lines with comon point (point 0) and with a given probability law $$p_ i$$ $$(i\leq k)$$ to select the $$i$$-th demi-line when the process attains 0 (for Lévy’s case: $$\mu=1/2$$, $$k=2$$, $$p_ 1=1/2$$). Let $$A_ i(t)$$, $$U_ i(t)$$ $$(i\leq k)$$ be occupation times on the $$i$$-th demi-line of the Walsh process and Walsh bridge before t, respectively, $$\ell_ t$$, $$\lambda_ t$$ be local times of the Bessel process and the Walsh bridge at 0 before $$t$$, respectively. Laws of $$\{A_ i(u)/(\ell_ u)^{1/\mu}$$ $$(i\leq k)\}$$ and $$\{A_ i(1)$$ $$(i\leq k)$$, $$(\ell_ 1)^{1/\mu}\}$$, expressions of $$E f(u_ i/(\lambda_ 1)^{1/\mu}$$ $$(i\leq k))$$ and $$E f(u_ i\;(i\leq k),(\lambda_ 1)^{-1/\mu})$$ are identified by laws and expectations of functions of $$k$$ independent stable variables. $$E f(A_ i(1)\;(i\leq k))$$, $$E f(u_ i\;(i\leq k))$$ and the Laplace transform of $$\{(t A_ i(t), (i\leq k))\}$$ are calculated explicitly. When $$p_ i\equiv 1/k\;(i\leq k)$$, the asymptotic law of $$\{k^{1/\mu}A_ i(t)\;(i\leq p),\;(\ell_ 1)^{1/\mu}\}$$ for fixed $$p$$ is also identified as $$k\to\infty$$.

##### MSC:
 60J65 Brownian motion 60J99 Markov processes