Domains of attraction to Tweedie distributions

A Convergence Theorem for Bivariate Exponential Dispersion Models

“…On the other hand, studies about convergence of EDMs performed by Jørgensen et al (1994) and Jørgensen et al (2009) allowed these authors to conjecture that similar results could be proved for those MEDMs they defined in 2013. Contributions made by J. Karamata are relevant for this type of convergence, he extended to k ℝ some Tauberian theorems for regular variation functions establishing that those functions behave asymptotically as their Laplace transforms.…”

“…On the other hand, Jørgensen (2013; Jørgensen and Martínez, 2013) developed a unified methodology to build Multivariate Exponential Dispersion Models (MEDMs) with fixed known marginals and a flexible correlation structure. Based on previous univariate studies about EDMs convergence (Jørgensen et al, 1994;2009) they could conjecture that previous results might be extended to those MEDMs that they had just defined. Moreover, MEDMs are important for practitioners because they broaden the parametric distribution family covered by GLMs derived from Multivariate Natural Exponential Family.…”

Limit theorems and phase transitions for two models of summation of i.i.d. random variables with a parameter

“…In [18] it is shown that elementary tempered stable distributions are the building blocks from which all other tempered stable distributions can be constructed. Such distributions have appeared in other limit theorems, see [11] and [26].…”

Taylor's law, via ratios, for some distributions with infinite mean

“…probability distributions, such as stable laws of index β with 0 < β < 1, which have infinite mean (for example, Jørgensen (1987Jørgensen ( ), (1997, Jørgensen et al (2009), Kendal (2004), (2013), and Kendal and Jørgensen (2011a), (2011b)). These studies sometimes assumed or imposed sequential dependence on observations, by contrast with our assumption that observations are independent.…”