  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.
