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A criterion of quasi-infinite divisibility for discrete laws

We consider arbitrary discrete probability laws on the real line. We obtain a criterion of their belonging to a new class of quasi-infinitely divisible laws, which is a wide natural extension of the class of well known infinitely divisible laws through the L\'evy type representations.


Introduction and problem setting
This note is devoted to the question posed in [1] about the description of quasi-infinitely divisibile laws within the class of arbitrary univariate discrete probability laws.
The class of quasi-infinitely divisible laws is a rather wide extention of the class of well known infinitely divisible laws. Let F be a distribution function on the real line with the characteristic function f . Following [12], F and f (and the corresponding probability law) are called quasi-infinitely divisible if f admits the following Lévy type representation: for some γ ∈ R (the set of real numbers), σ 2 0, and function Λ (the Lévy spectral function), which has a finite total variation on every interval (−∞, −r] and [r, ∞), r > 0, and it satisfies Here f uniquely determines the triplet (γ, σ 2 , Λ). Quasi-infinitely divisible F and f are rationally infinitely divisible, i.e. there exist infinitely divisible distribution functions F 1 and F 2 with characteristic functions f 1 and f 2 , respectively, such that F 1 = F * F 2 , where " * " is the convolution, or in the equivalent form: It is clear that every infinitely divisible distribution function is quasi-infinitely divisible with non-decreasing Λ on every interval (−∞, 0) and (0, ∞). The opposite is not true, the examples can be found in the classical monographs [7], [14] and [15]. The first detailed analysis of quasi-infinitely divisible laws based on their Lévy type representations was performed in [12], where the authors studied questions concerning supports, moments, continuity, and weak convergence for these laws. These results were generalized and complemented in the papers [1], [2], [3], [4], [5], [8], [9] and [10]. The quasi-infinitely divisible laws have already found a lot of interesting applications (see [5], [6], [13], [16], [17] and the references given there).
It is rather natural and interesting to investigate criteria of quasi-infinite divisibility for probability laws. The most deep results here were obtained for univariate discrete laws in [1], [9], and [12]. In particular, following [12], a discrete lattice probability law is quasi-infinitely divisible if its characteristic function f has no zeroes on the real line, i.e. f (t) = 0, t ∈ R. For arbitrary discrete laws only the sufficient condition is known due to [1]. Namely, if f is separated from zero, i.e. |f (t)| µ > 0 for some µ and all t ∈ R, then the law is quasi-infinitely divisible (see the next section for more details). The latter result generalizes the previous, because in the lattice case the function t → |f (t)|, t ∈ R, is continuous and periodic, and the absence of zeroes is equivelent to the separatness from zero over the period segment (see [1] for more details). Anyway, however, the following interesting question remains here (it was posed in [1]). Are all characteristic functions of quasi-infinitely divisible laws separated from zero? In this note we will show that it is true. So we obtain a criterion of quasi-infinite divisibility for arbitrary discrete laws.
We will use the following notation. We denote by R, Z and N the sets of real numbers, integers, and positive integers, respectively. We write a n ∼ b n for number sequences a n and b n , n ∈ N, if a n /b n → 1, n → ∞.

The result
Let us consider an arbitrary discrete distribution function where x k , k ∈ N, are distinct real numbers, p x k 0, k ∈ N, and ∞ k=1 p x k = 1. Let f be the characteristic function of F : Let X be the set of all points of growth of F , i.e. X := {x k : p x k > 0, k ∈ N} = ∅. Let us introduce the set of all finite Z-linear combinations of elements from the set X: In other words, X is a module over the ring Z with the generating set X. Since X = ∅, X is an infinite countable set. The following result was obtained in [1].
Theorem 1 Suppose that inf t∈R |f (t)| > 0, i.e. there exists µ > 0 such that |f (t)| µ > 0 for all t ∈ R. Then f admits the following representation where γ 0 ∈ X , λ u ∈ R for all u ∈ X \ {0}, and u∈ X \{0} |λ u | < ∞. Here F is quasi-infinitely divisible and f has representation (1) with Thus the condition inf t∈R |f (t)| > 0 implies the quasi-infinite divisibility of F . We complement this fact by the following proposition.
Proof. It is known that characteristic functions of quasi-infinitely divisible laws have no zeroes on the real line (see [12]). Therefore if there exists t 0 ∈ R such that f (t 0 ) = 0, then F is not quasi-infinitely divisible. Hence we focus only on the case f (t) = 0 for all t ∈ R with inf t∈R |f (t)| = 0.
Suppose, contrary to our claim, that F is a quasi-infinitely divisible distribution function. Then f admits representation (1) with some triplet (γ, σ 2 , Λ). Let us consider for fixed τ ∈ R the following function Due to the conditions for Λ, for any fixed τ ∈ R we have Let (t m ) m∈N be an increasing sequence such that t m → +∞ and f (t m ) → 0, m → ∞. Let us define ϕ m (τ ) := f (t m + τ ), τ ∈ R, m ∈ N. Since f is an almost periodic function by (2), the sequence (ϕ m ) m∈N is relatively compact in the topology of uniform convergence on R (see [11] pp. 23-24). So there exists a subsequence (ϕ m l ) l∈N that uniformly converges to an almost periodic function ϕ, i.e.
In particular, for any fixed s ∈ R we have Since f (t) = 0, t ∈ R, we conclude that |f (2t m l + s)| → 0, l → ∞ for any fixed s ∈ R. Next, from the sequence (m l ) l∈N we can choose a subsequence (m ′ l ) l∈N such that the functions s → f (2t m ′ l + s) uniformly converge on R as l → ∞ (see [11] pp. 23-24). It is clear that their uniform limit is zero, i.e. we have So, in particular, |f (0)| = |f (2t m ′ l − 2t m ′ l )| → 0, l → ∞, but f (0) = 1, a contradiction. Thus there exists τ ∈ R such that ϕ(τ )ϕ(−τ ) = 0. Then, due to (6) and the convergence f (t m l ) → 0, l → ∞, for this τ we have Thus we found τ such that sup t∈R |ψ τ (t)| = ∞. This contradicts (4). Therefore f can not be the characteristic function of a quasi-infinitely divisible law. ✷ The examples of f satisfying inf t∈R |f (t)| = 0 were considered in [1] and [9]. Thus Theorem 1 and Theorem 2 yield the following criterion: discrete distribution function F is quasi-infinitely divisible if and only if inf t∈R |f (t)| > 0, i.e. there exists µ > 0 such that |f (t)| µ > 0 for all t ∈ R. Moreover, now we know that the characteristic functions of all discrete quasi-infinitely divisible laws admit representation (3) with some γ 0 and λ u .

Acknowledgments
The author is supported by the Ministry of Science and Higher Education of the Russian Federation, agreement 075-15-2019-1620 date 08/11/2019.