Some definite integrals arising from selfdecomposable characteristic functions

2023/01/27 by Jurek, Zbigniew J.
#2020 Mathematics Subject Classifications: Primary: 60E07 #60E10 #60H05 #FOS: Mathematics #Probability (math.PR) #Secondary: 60G51

paper · doi:10.48550/arxiv.2301.11625

Abstract

In the probability theory selfdecomposable, or class L₀ distributions play an important role as they are limiting distributions of normalized partial sums of sequences of independent, not necessarily identically distributed, random variables. The class L₀ is quite large and includes many known classical distributions and statistics. For this note the most important feature of the selfdecomposable variables are their random integral representation with respect to Lévy process. From those random integral representation we get equality of logarithms of some characteristic functions. These allows us to get formulas for some definite integrals, some of them probably were unknown before.

Related

• A new factorization property of the selfdecomposable probability measures
• Classes of infinitely divisible distributions on R^d related to the class of selfdecomposable distributions
• Stochastic integral characterizations of semi-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes
• Some characterizations of multiple selfdecomposability with extensions and an application to the Gamma function
• Selfdecomposable Fields