On Convergence and Compactness in Variation with a Shift of Discrete Probability Laws

On weak convergence of quasi-infinitely divisible laws

“…We use u → 1 τ sin(τ u) as the centering function in the integral in (1) following Zolotarev [33; 34]. If (1) holds for some τ = τ 0 > 0, then it holds for any τ > 0, where γ will depend on τ , but G will not. It is well known that the spectral pair (γ , G) is uniquely determined by f and hence by F. The Lévy-Khinchin formula plays a fundamental role in probability theory; it also has a lot of applications in related fields (see [3; 30; 31]).…”

“…This class of so-called quasi-infinitely divisible laws was introduced by Lindner and Sato [22]. Following them, a distribution function F (and the corresponding law) is called quasi-infinitely divisible if its characteristic function f admits the representation (1) with some shift parameter γ ∈ ‫,ޒ‬ spectral function G : ‫ޒ‬ → ‫ޒ‬ of bounded variation on ‫ޒ‬ (not necessarily monotone), and for some (any) τ > 0. Here G is assumed to be right-continuous with condition G(−∞) = 0 as before and so f (and F) uniquely determines the spectral pair (γ , G) (see [13, p. 80]).…”

“…Namely, let (F n ) n∈‫ގ‬ be a sequence of quasi-infinitely divisible distribution functions and let (γ n , G n ) be the spectral pair of F n for every n ∈ ‫.ގ‬ Let F be a quasi-infinitely divisible distribution function with the spectral pair (γ , G). Then the results from [23] are in fact the following: (1)…”

On weak convergence of quasi-infinitely divisible laws

“…Various forms of definition and the first detailed analysis of the class of quasi-infinitely divisible laws on R was performed in [22], the multivariate case is considered in the recent papers [6], [7], and [21]. There are some results for discrete probability laws in this field (see [1], [2], [17], [18], and [19]) and for mixed laws (see [4] and [5]). It should be noted that quasi-infinitely divisible laws now have interesting applications in theory of stochastic processes (see [23] and [28]), number theory (see [26]), physics (see [11] and [12]), and insurance mathematics (see [31]).…”