Abstract
In the first short survey article [H.-J. Rossberg,Wiss. Z. Karl Marx Univ.,37, 366–374 (1988)] the author indicated that positive definite probability densities deserve attention and that the investigation of this class leads to serious new mathematical problems. Later [R. Riedel and H.-J. Rossberg,Metrika, (submitted for publication)] it turned out that it is worthwhile to consider an even larger class of distribution functions, namely those having characteristic function f≥0; they were first briefly considered in [P. Lévy,C. R. Acad. Sci.,265, 249–252 (1967)]. We survey the basic ideas and the present state of the developing new theory; moreover, we formulate open problems. Some new results are proved. For the sake of brevity we do not enter into a discussion of continuation problems and limit theorems; for these subjects see [H.-J. Rossberg,Appendix to Gnedenko B. W., Einführung in die Wahrscheinlichkeitstheorie, 9th ed., Akademie-Verlag, Berlin (1991)], [H.-J. Rossberg,Math. Nachr.,141, 227–232 (1989)], and [H.-J. Rossberg,Lect. Notes Math.,1412, 296–308 (1989)].
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Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.
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Rossberg, H.J. Positive definite probability densities and probability distributions. J Math Sci 76, 2181–2197 (1995). https://doi.org/10.1007/BF02363232
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DOI: https://doi.org/10.1007/BF02363232