Abstract
This chapter examines Lévy processes (LPs). These processes can be thought of as random walks in continuous time, having independent and stationary increments, with the assumption that the characteristic function of the state of an LP at time 1 satisfies a certain “Standard Condition” (SC). Exponential Lévy processes (ELPs) in particular are quite attractive for modeling many phenomena. The chapter begins by explaining LPs in more detail, introducing the basic notions as well as providing examples of Lévy processes. It then uses a certain variant of the Poisson Summation Formula to arrive at convergence results for the expectations of certain normalized functionals of the significand of an ELP and obtain, using Azuma's inequality for martingales, large deviation results for these functionals. From here, the chapter calculates the a.s. (almost surely) convergence of normalized functionals and discusses further related conditions and theorems.
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