A note on the gaps in the support of discretely infinitely divisible laws. (English) Zbl 1302.60036
Summary: We discuss the nature of gaps in the support of a discretely infinitely divisible distribution from the angle of compound Poisson laws/processes. The discussion is extended to infinitely divisible distributions on the nonnegative real line.
MSC:
| 60E07 | Infinitely divisible distributions; stable distributions |
References:
| [1] | A. Bose, A. Dasgupta, and H. Rubin, “A contemporary review and bibliography of infinitely divisible distributions and processes,” Sankhy\Ba A, vol. 64, no. 3, pp. 763-819, 2002. · Zbl 1192.60036 |
| [2] | S. Satheesh, “Why there are no gaps in the support of non-negative integer-valued infinitely divisible laws?” ProbStat Models, vol. 3, pp. 1-7, 2004, http://arxiv.org/abs/math/0409042. |
| [3] | J. Grandell, Mixed Poisson Processes, vol. 77, Chapman & Hall, London, UK, 1997. · Zbl 0922.60005 |
| [4] | F. W. Steutel and K. van Harn, Infinite Divisibility of Probability Distributions on the Real Line, Marcel Dekker, New York, NY, USA, 2004. · Zbl 1063.60001 |
| [5] | A. G. Pakes, J. Navarro, J. M. Ruiz, and Y. del Aguila, “Characterizations using weighted distributions,” Journal of Statistical Planning and Inference, vol. 116, no. 2, pp. 389-420, 2003. · Zbl 1021.62004 · doi:10.1016/S0378-3758(02)00357-9 |
| [6] | E. Seneta, Non-Negative Matrices and Markov Chains, Springer, New York, NY, USA, 2nd edition, 1981. · Zbl 0471.60001 |
| [7] | E. Seneta and D. Vere-Jones, “On a problem of M. Jiřina concerning continuous state branching processes,” Czechoslovak Mathematical Journal, vol. 19, pp. 277-283, 1969. · Zbl 0181.44002 |
| [8] | M. Loève, Probability Theory. I, Springer, New York, NY, USA, 4th edition, 1977. |
| [9] | H. G. Tucker, “The supports of infinitely divisible distribution functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 436-440, 1975. · Zbl 0308.60014 · doi:10.2307/2040661 |
| [10] | S. Orey, “On continuity properties of infinitely divisible distribution functions,” Annals of Mathematical Statistics, vol. 39, pp. 936-937, 1968. · Zbl 0172.22101 · doi:10.1214/aoms/1177698325 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
