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

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. 1. Introduction We begin with the following definitions in which denotes the set of positive integers and the set of nonnegative integers. Definition 1. A -valued random variable (r.v) is infinitely divisible (ID) if for every there exists a r.v such that Definition 2. A -valued r.v is discretely infinitely divisible (DID) if for every there exists a -valued r.v such that (1) is true. The distinction between the notions of infinite divisibility (ID) of discrete distributions in general and discrete infinite divisibility (DID) is often not made and this sometimes causes confusion. For example, the geometric distribution on is ID but not DID whereas that on is DID and thus is ID. Blurring these concepts may have lead to Remark 9 in Bose et al. [1] asserting that if a -valued ID distribution assigns a positive probability to the integer 1, then its support cannot have any gaps. Satheesh [2] gave the following simple example to show that this is not always true. Example 3. Consider the r.v with probability generating function (PGF) , integer. Obviously, , but its support has gaps as the atoms are separated by integers. In the sequel or denotes the support of or that of its distribution function (d.f.) . To emphasize what now is obvious, if is ID and is its left extremity and if , then is DID and its support has gaps if . Also, in terms of Definition 1, , and hence only if is DID. To avoid such cases, unless otherwise stated, we assume that is aperiodic; that is, its greatest common divisor is unity. This natural restriction implies the following definition. Definition 4. The support of the ID discrete r.v. on contains gaps if it is a proper subset of . (Note that is necessarily an infinite set). The distinction between Definitions 1 and 2 is made clear in Grandell [3, page 26; a result due to Kallenberg], Satheesh [2], and Steutel and van Harn [4, page 23]. Implications of the r.v being DID include the following.(i) .(ii)Let be the th convolution root of ; that is, is the d.f. of in (1). Then or in terms of their d.f.’s = for all .(iii)Let and denote the PGF’s of and , respectively. Then , , for every , that is, for all ; the root of a DID PGF is again a PGF.(iv) is compound Poisson; that is, , where is a PGF and the rate parameter . We conclude this review by noting that Satheesh [2] modified Remark 9 in Bose et al. [1]