Showing papers on "Subordinator published in 1994"
01 Jan 1994
TL;DR: In this paper, the Ornstein-Uhlenbeck process X(t)=e−t/2Z(et) was introduced to show that the number of windings made by a planar Brownian motion Z around the origin and taken in the logarithmic time-scale is asymptotically close to a Cauchy process.
Abstract: A celebrated theorem of Spitzer suggests that the number of windings made by a planar Brownian motion Z around the origin and taken in the logarithmic time-scale, is asymptotically close to a Cauchy process. The purpose of this paper is to show that this informal consideration can be made precise by introducing the Ornstein-Uhlenbeck process X(t)=e−t/2Z(et). This yields short proofs of known results as well as some new features on the asymptotic behaviour of the winding number (in distribution and pathwise).
37 citations
TL;DR: In this paper, a theory of semiregenerative phenomena was developed by the author, and the set of points at which such a phenomenon occurs is called a semi regenerative set.
Abstract: A theory of semiregenerative phenomena was developed by the author. The set of points at which such a phenomenon occurs is called a semi regenerative set. There is a correspondence between a semiregenerative set and the range of a Markov subordinator with a unit drift (or a Markov renewal process in the discrete time case). Prabhu, Tang, and Zhu showed that the properties of semiregenerative sets associated with Markov random walks completely characterize the fluctuation behaviour of these processes in the nondegenerate case and also established a Wiener-Hopf factorization based on these sets. These results are surveyed in this paper.