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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
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