ON PAUL LÉVY’S ARC SINE LAW AND SHIGA-WATANABE’S TIMEINVERSION RESULT
Svend Erik Graversen Juha Vuolle-Apiala
Abstract: Let be a symmetric real-valued -self-similar diffusion starting at
We characterize the distributions of the time spent on before time
and the time of the last visit to before This gives a simple new proof
to well-known results including P. Lévy’s arc sine law for Brownian motion and
Brownian bridge and similar results for symmetrized Bessel processes. Our focus
is more on simplicity of proofs than on novelty of results. Section contains a
generalization of T. Shiga’s and S. Watanabe’s theorem on time inversion for Bessel
processes. We show that their result holds also for symmetrized Bessel processes.
be a symmetric real-valued 
-self-similar diffusion starting at 
We characterize the distributions of 
the time spent on 
before time 
and 
the time of the last visit to 
before 
This gives a simple new proof
to well-known results including P. Lévy’s arc sine law for Brownian motion and
Brownian bridge and similar results for symmetrized Bessel processes. Our focus
is more on simplicity of proofs than on novelty of results. Section 
contains a
generalization of T. Shiga’s and S. Watanabe’s theorem on time inversion for Bessel
processes. We show that their result holds also for symmetrized Bessel processes.