Abstract
A great deal of work has been done during the last 45 years concerning the unimodality of one-dimensional infinitely divisible distribution functions. Recently, a few results have been obtained for multivariate infinitely divisible distribution functions. The purpose of this paper is to give a survey of previous work and to discuss some unsolved problems.
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References
Anderson, T. W. (1955) The interval of a symmetric unimodal function, Proc. Amer. Math. Soc. 6, 170–176.
Chung, K. L. (1953) Sur les lois des probabilities unimodales, C. R. Acad. Sci. Paris 236, 583–584.
Dharmadhikari, S. W. and Jogdeo, K. (1976) Multivariate unimodality, Ann. Statist. 4, 607–613.
Gnedenko, B. V. and Kolmogorov, A. N. (1954) Limit Distributions for Sums of Independent Random Variables (translation K. L. Chung) Reading, Mass., Addison Wesley.
Ibraginov, I. A. (1957) A remark on probability distributions of class L, Theor. Probability Appl. 2, 117–119.
Ibraginov, I. A. and Chernin, K. F. (1959) On the unimodality of stable laws, Theor. Probability Appl. 4, 417–419.
Kanter, M. (1976) On the unimodality of stable densities, Ann. Probability 4, 1006–1008.
Kanter, M. (1977) Unimodality and dominance of symmetric random vectors, Trans. Amer. Math. Soc. 229, 65–86.
Lapin, A. I. (1947) On properties of stable laws, Ph.D. dissertation.
Medgyessy, P. (1967) On a new class of unimodal infinitely divisible distribution functions and related topics, Studia Sci. Math. Hungar. 2, 441–446.
Olshen, R. A. and Savage, L. J. (1970) A generalized unimodality, J. Appl. Prob. 7, 21–34.
Sato, K. and Yamazato, M. (1978) On distribution functions of class L, Z. Wahrschienlichkeitstheorie verw. Gebiete 43, 273–308.
Sharpe, M. (1969) Stable distributions on vector groups, Trans. Amer. Math. Soc. 136, 119–148.
Sherman, S. (1955) A theorem on convex sets with applications, Ann. Math. Statist. 26, 763–767.
Steutel, F. and van Harn, K. (1979) Discrete analogues of self-decomposibility and stability, Ann. Probability 7, 893–899.
Sun, T. C. (1967) A note on the unimodality of distributions of class L, Ann. Math. Statist. 38, 1296–1299.
Urbanik, K. (1972) Levy’s probability measures on Euclidean spaces, Studia Math. 44, 119–148.
Wintner, A. (1936) On a class of Fourier transforms, Amer. J. Math. 58, 45–90.
Wintner, A. (1956) Cauchy’s stable distributions and an "explicit formula" of Mellin, Amer. J. Math. 78, 319–361.
Wells, D. R. (1978) A monotone unimodal distribution which is not central convex unimodal, Ann. Statist. 6, 926–931.
Wolfe, S. J. (1971) On the unimodality of L functions, Ann. Math. Statist. 42, 912–918.
Wolfe, S. J. (1978) On the unimodality of symmetric distribution functions of class L, J. Multivar. Anal. 5, 236–242.
Wolfe, S. J. (1978) On the unimodality of infinitely divisible distribution functions, Z. Wahrscheinlichkeitstheorie verw. Gebiete 45, 229–235.
Wolfe, S. J. (1980) A characterization of Lévy probability distribution functions on Euclidean spaces, J. Multivar. Ann. 10 379–384
Yamazato, M. (1978) Unimodality of infinitely divisible distribution functions of class L, Ann. Probability 6, 573–531.
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© 1981 Springer-Verlag
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Wolfe, S.J. (1981). On the unimodality of infinitely divisible distribution functions II. In: Dugué, D., Lukacs, E., Rohatgi, V.K. (eds) Analytical Methods in Probability Theory. Lecture Notes in Mathematics, vol 861. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097324
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DOI: https://doi.org/10.1007/BFb0097324
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