Stable random processes and stochastic integrals

An α-stable stochastic process is a random element whose finite-dimensional distributions are α-stable. It is used to introduce the notion of α-stable stochastic integrals. It is convenient to view these integrals as α-stable stochastic processes parameterized by their integrands. This chapter develops some basic properties of stable integrals. The representation theorem states that an α-stable random vector can be represented as an α-stable stochastic integral. The representation sheds light into the structure of α-stable stochastic processes. The chapter describes that the series representation for one-dimensional α-stable random variables can be extended to α-stable random measures, and it indicates that α-stable random measures have essentially a "discrete" structure.