×

zbMATH — the first resource for mathematics

Generalized arc-sine laws for one-dimensional diffusion processes and random walks. (English) Zbl 0824.60080
Cranston, Michael C. (ed.) et al., Stochastic analysis. Proceedings of the Summer Research Institute on stochastic analysis, held at Cornell University, Ithaca, NY, USA, July 11-30, 1993. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 57, 157-172 (1995).
Let \(m : (- \infty, \infty) \to [- \infty, \infty]\) be a right-continuous non-decreasing function, and \(X\) a one-dimensional (generalized) diffusion process in natural scale and with speed measure \(dm\). So, when \(m\) is smooth, the infinitesimal generator of \(X\) is \(d^ 2x/dm (x)dx\). Consider the time spent by the diffusion in \([0, \infty)\), \(A_ t = \int^ t_ 0 \mathbf{1}_{\{X_ S \geq 0 \}} ds\), \(t > 0\). The purpose of the paper is to decide whether \(t^{-1} A_ t\) converges in distribution as \(t \to \infty\), and then to determine the limiting law. Typically, \(t^{-1} A_ t\) converges to a nondegenerate limit if and only if \(m\) is regularly varying at \(+\infty\) with a positive index and \(\lim_{x \to \infty} m(x)/m (-x) = c \in (0, \infty)\). The limiting distribution is then a generalized arcsine law introduced by Lamperti. The approach relies on M. G. Krein’s spectral theory of vibrating strings. Analogous results hold for space-dependent random walks.
For the entire collection see [Zbl 0814.00017].
Reviewer: J.Bertoin (Paris)

MSC:
60J60 Diffusion processes
60J55 Local time and additive functionals
PDF BibTeX XML Cite