Location of the Path Supremum for Self-similar Processes with Stationary Increments

TL;DR

This paper investigates the distribution of the location of the maximum of self-similar processes with stationary increments, providing a spectral-type representation and bounds, with applications to Lévy processes.

Contribution

It introduces a point process framework to analyze the path supremum location distribution, revealing its mixture structure and establishing bounds on its density.

Findings

Distribution has a spectral-type mixture form

Bounds on density and derivatives are established

Results extend to various random location problems

Abstract

In this paper we consider the distribution of the location of the path supremum in a fixed interval for self-similar processes with stationary increments. To this end, a point process is constructed and its relation to the distribution of the location of the path supremum is studied. Using this framework, we show that the distribution has a spectral-type representation, in the sense that it is always a mixture of a special group of absolutely continuous distributions, plus point masses on the two boundaries. Bounds on the value and the derivatives of the density function are established. We further discuss self-similar L\'{e}vy processes as an example. Most of the results in this paper can be generalized to a group of random locations, including the location of the largest jump, etc.