A Note on the Gaps in the Support of Discretely Infinitely Divisible Laws

Anthony G Pakes,S Satheesh

doi:10.1155/2013/695623

Abstract

We discuss the nature of gaps in the support of a discretely infinitely divisible distribution from the angle of compound Poisson laws/processes. The discussion is extended to infinitely divisible distributions on the nonnegative real line.

Highlights

We begin with the following definitions in which N denotes the set of positive integers and Z+ the set of nonnegative integers.Definition 1
The geometric distribution on {1, 2, . . .} is infinitely divisible (ID) but not discretely infinitely divisible (DID) whereas that on Z+ = {0, 1, 2, . . .} is DID and is ID. Blurring these concepts may have lead to Remark 9 in Bose et al [1] asserting that if a Z+-valued ID distribution assigns a positive probability to the integer 1, its support cannot have any gaps
Unless otherwise stated, we assume that supp(X − lX) is aperiodic; that is, its greatest common divisor is unity

Summary

Introduction

We begin with the following definitions in which N denotes the set of positive integers and Z+ the set of nonnegative integers. Blurring these concepts may have lead to Remark 9 in Bose et al [1] asserting that if a Z+-valued ID distribution assigns a positive probability to the integer 1, its support cannot have any gaps. In terms of Definition 1, lXn = lX/n, and supp(Xn) = supp(X) only if X is DID To avoid such cases, unless otherwise stated, we assume that supp(X − lX) is aperiodic; that is, its greatest common divisor is unity. Unless otherwise stated, we assume that supp(X − lX) is aperiodic; that is, its greatest common divisor is unity This natural restriction implies the following definition. The purpose of this note is to complement the discussion on the gaps in the support of DID laws in Satheesh [2] from the angle of compound Poisson laws/processes.

Compound Poisson Processes and DID

Some General Considerations