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We consider the class of quasi-infinitely divisible distributions. These distributions have appeared before in the theory of decompositions of probability laws, and nowadays they have various applications in theory of stochastic processes, physics, and insurance mathematics. The characteristic functions of quasi-infinitely divisible distributions admit Lévy type representation with real drift, nonnegative Gaussian variance, and "signed Lévy measure". Lindner et al. (2018) have recently done the first detailed analysis of these distributions based on such representations. The most complete results were established by them for the quasi-infinitely divisible distributions on the integers. In particular, the authors have obtained a criterion of weak convergence for distributions from this class in terms of parameters of their Lévy type representations. In the present short paper we complement this result by similar criteria of relative and stochastic compactness for quasi-infinitely divisible distributions with partial weak limits from this class. We also show that if a general sequence of distributions on the integers is relatively compact with quasi-infinitely divisible partial weak limits, then all distributions of the sequence are quasi-infinitely divisible except a finite number. [ABSTRACT FROM AUTHOR]

Abstract: Mukherjea et al. [Mukherjea, A., Rao, M., Suen, S., 2006. A note on moment generating functions. Statist. Probab. Lett. 76, 1185–1189] proved that if a sequence of moment generating functions converges pointwise to a moment generating function for all in some open interval of the real line, not necessarily containing the origin, then the distribution functions (corresponding to ) converge weakly to the distribution function (corresponding to ). In this note, we improve this result and obtain conditions of the convergence which seem to be sharp: converge weakly to if converge to , , for some sequence having the minimal and the maximal points. A similar result holds for characteristic functions. [Copyright &y& Elsevier]

Abstract: Let , be a triangular array of independent elliptical random vectors in . In this paper we investigate the asymptotic behaviour of the multivariate maxima of this triangular array. Generalising previous results for the bivariate set-up, we show that the normalised maxima of this elliptical triangular array is attracted by the multivariate Hüsler–Reiss distribution function provided that the components of the triangular array become asymptotically dependent with a specific rate, and further the random radius pertaining to the elliptical random vectors is in the Gumbel max-domain of attraction. [Copyright &y& Elsevier]

Abstract: Let be a triangular array of independent bivariate elliptical random vectors. Hüsler and Reiss (1989. Statist. Probab. Lett. 7, 283–286) show that for the particular case that the array is Gaussian, the maxima of this array is in the max-domain of attraction of Hüsler–Reiss distribution function, provided that an asymptotic condition holds for the correlation . In this paper we obtain a similar result for the more general case of elliptical triangular arrays. [Copyright &y& Elsevier]

In this paper, we study the asymptotic distribution of the generally normalized maximum under finite mixture models. In one of the two theorems presented, we obtain a necessary and sufficient condition for this weak convergence, as well as the limit forms. The second theorem gives sufficient conditions for this convergence when the components of the mixture have different general normalizations. Examples are given to illustrate the applications of the two theorems. [Copyright &y& Elsevier]