×

A Cramér-Wold device for infinite divisibility of \(\mathbb{Z}^d \)-valued distributions. (English) Zbl 1527.60019

The distribution of a \(\mathbb{Z}^d \)-valued random vector \(X\) is infinitely divisible if and only if the distribution of \(a^{T}X\) is infinitely divisible for all \(a\in \mathbb R^{d}\), as shown by the proof of the validity of a Cramér-Wold device for \(\mathbb{Z}^d \)-valued distributions. This is equivalent to the distribution of \(a^{T}X\) being infinitely divisible for all \(a\in \mathbb N_0^{d}\). If the characteristic function of a \(\mathbb{Z}^d \)-valued distribution is zero-free, a Lévy-Khintchine type representation with a signed Lévy measure is a crucial tool for demonstrating this.

MSC:

60E07 Infinitely divisible distributions; stable distributions
60G51 Processes with independent increments; Lévy processes
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Alexeev, I.A. and Khartov, A.A. (2021). Spectral representations of characteristic functions of discrete probability laws. Preprint. Available at arXiv:2101.06038. · Zbl 1476.60034
[2] Basrak, B., Davis, R.A. and Mikosch, T. (2002). A characterization of multivariate regular variation. Ann. Appl. Probab. 12 908-920. · Zbl 1070.60011 · doi:10.1214/aoap/1031863174
[3] Berger, D. (2019). On quasi-infinitely divisible distributions with a point mass. Math. Nachr. 292 1674-1684. · Zbl 1480.60038 · doi:10.1002/mana.201800073
[4] Calderón, A., Spitzer, F. and Widom, H. (1959). Inversion of Toeplitz matrices. Illinois J. Math. 3 490-498. · Zbl 0091.11101
[5] Chhaiba, H., Demni, N. and Mouayn, Z. (2016). Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels. J. Math. Phys. 57 072103, 14 pp. · Zbl 1342.81106 · doi:10.1063/1.4958724
[6] Cramér, H. and Wold, H. (1936). Some Theorems on Distribution Functions. J. Lond. Math. Soc. 11 290-294. · Zbl 0015.16801 · doi:10.1112/jlms/s1-11.4.290
[7] Cuppens, R. (1975). Decomposition of Multivariate Probabilities. Probability and Mathematical Statistics, Vol. 29. New York: Academic Press. · Zbl 0363.60012
[8] Demni, N. and Mouayn, Z. (2015). Analysis of generalized Poisson distributions associated with higher Landau levels. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 1550028, 13 pp. · Zbl 1333.81454 · doi:10.1142/S0219025715500289
[9] Dwass, M. and Teicher, H. (1957). On infinitely divisible random vectors. Ann. Math. Stat. 28 461-470. · Zbl 0078.31303 · doi:10.1214/aoms/1177706974
[10] Grafakos, L. (2014). Classical Fourier Analysis, 3rd ed. Graduate Texts in Mathematics 249. New York: Springer. · Zbl 1304.42001
[11] Hult, H. and Lindskog, F. (2006). On Kesten’s counterexample to the Cramér-Wold device for regular variation. Bernoulli 12 133-142. · Zbl 1108.60015
[12] Ibragimov, I.A. (1972). On a problem of C. R. Rao on I. D. laws. Sankhyā Ser. A 34 447-448. · Zbl 0262.60004
[13] Khartov, A.A. (2019). Compactness criteria for quasi-infinitely divisible distributions on the integers. Statist. Probab. Lett. 153 1-6. · Zbl 1458.60023 · doi:10.1016/j.spl.2019.05.008
[14] Lindner, A., Pan, L. and Sato, K. (2018). On quasi-infinitely divisible distributions. Trans. Amer. Math. Soc. 370 8483-8520. · Zbl 1428.60034 · doi:10.1090/tran/7249
[15] Linnik, Y.V. (1964). Decomposition of Probability Distributions. Edinburgh: Oliver and Boyd Ltd. · Zbl 0121.35502
[16] Nakamura, T. (2015). A complete Riemann zeta distribution and the Riemann hypothesis. Bernoulli 21 604-617. · Zbl 1328.60048 · doi:10.3150/13-BEJ581
[17] Passeggeri, R. (2020). Spectral representations of quasi-infinitely divisible processes. Stochastic Process. Appl. 130 1735-1791. · Zbl 1471.60073 · doi:10.1016/j.spa.2019.05.014
[18] Rudin, W. (1962). Fourier Analysis on Groups. Interscience Tracts in Pure and Applied Mathematics, No. 12. New York: Wiley. · Zbl 0107.09603
[19] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Stochastic Modeling. New York: CRC Press. · Zbl 0925.60027
[20] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press. · Zbl 0973.60001
[21] Zhang, H., Liu, Y. and Li, B. (2014). Notes on discrete compound Poisson model with applications to risk theory. Insurance Math. Econom. 59 325-336 · Zbl 1306.60050 · doi:10.1016/j.insmatheco.2014.09.012
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.