On convergence and compactness in variation with shift of discrete probability laws
DOI:
https://doi.org/10.21638/spbu01.2021.301Abstract
We consider a class of discrete distribution functions, whose characteristic functions are separated from zero, i. e. their absolute values are greater than positive constant on the real line. The class is rather wide, because it contains discrete infinitely divisible distribution functions, functions of lattice distributions, whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently the authors showed that characteristic functions of elements of this class admit the L´evy-Khinchine type representations with non-monotonic spectral function. Thus our class is included in the set of so called quasi-infinitely divisible distribution functions. Using these representation the authors also obtained limit and compactness theorems with convergence in variation for the sequences from this class. This note is devoted to similar results concerning convergence and compactness but with weakened convergence in variation. Replacing of type of convergence notably expands applicability of the results.Keywords:
characteristic functions, L´evy-Khinchine type representations, quasi-infinitely divisible distributions, convergence in variation, relative compactness, entropy, uncertainty coefficient, iterative procedure, symptom-syndromic method, dimension reduction, classification, sensitivity, specificity
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References
Литература
1. Alexeev I.A., Khartov A.A. Spectral representations of characteristic functions of discrete probability laws. arXiv:2101.06038 (2021).
2. Lindner A., Pan L., Sato K. On quasi-infinitely divisible distributions. Trans. of AMS 370, 8483–8520 (2018).
3. Berger D., Lindner A., A Cram´er Wold device for infinite divisibility of Zd-valued distributions. arXiv:2011.08530 (2020).
4. Хартов А.А., Алексеев И.А. Квази-безграничная делимость и трехточечные вероятностные законы. Записки науч. сем. ПОМИ 495, 305–316 (2020).
5. Lindner A., Sato K. Properties of stationary distributions of a sequence of generalized Ornstein Uhlenbeck processes. Math. Nachr. 284 (17–18), 2225–2248 (2011).
6. Berger D. On quasi-infinitely divisible distributions with a point mass. Math. Nachr. 292, 1674–1684 (2018).
7. Khartov A.A. Compactness criteria for quasi-infinitely divisible distributions on the integers. Stat. & Probab. Letters 153, 1–6 (2019).
8. Berger D., Kutlu M., Lindner A. On multivariate quasi-infinitely divisible distributions. arXiv:2101.02544 (2021).
9. Passeggeri P. Spectral representation of quasi-infinitely divisible processes. Stoch. Process. Appl. 130, iss. 3, 1735–1791 (2020).
10. Nakamura T. A complete Riemann zeta distribution and the Riemann hypothesis. Bernoulli 21, iss. 3, 1735–1791 (2015).
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References
1. Alexeev I.A., Khartov A.A. Spectral representations of characteristic functions of discrete probability laws. arXiv:2101.06038 (2021).
2. Lindner A., Pan L., Sato K. On quasi-infinitely divisible distributions. Trans. of AMS 370, 8483–8520 (2018).
3. Berger D., Lindner A., A Cram´er Wold device for infinite divisibility of Zd-valued distributions. arXiv:2011.08530 (2020).
4. Khartov A.A., Alexeev I.A. Quasi-infinite divisibility and three-point probability laws. Zapiski Nauchnykh Seminarov POMI 495, 305–316 (2020). (In Russian)
5. Lindner A., Sato K. Properties of stationary distributions of a sequence of generalized Ornstein Uhlenbeck processes. Math. Nachr. 284 (17–18), 2225–2248 (2011).
6. Berger D. On quasi-infinitely divisible distributions with a point mass. Math. Nachr. 292, 1674–1684 (2018).
7. Khartov A.A. Compactness criteria for quasi-infinitely divisible distributions on the integers. Stat. & Probab. Letters 153, 1–6 (2019).
8. Berger D., Kutlu M., Lindner A. On multivariate quasi-infinitely divisible distributions. arXiv:2101.02544 (2021).
9. Passeggeri P. Spectral representation of quasi-infinitely divisible processes. Stoch. Process. Appl. 130, iss. 3, 1735–1791 (2020).
10. Nakamura T. A complete Riemann zeta distribution and the Riemann hypothesis. Bernoulli 21, iss. 3, 1735–1791 (2015).
11. Chhaiba H., Demni N., Mouayn Z. Analysis of generalized negative binomial distributions attached to hyperbolic Landau levels. J. Math. Phys. 57, 072103 (2016). https://doi.org/10.1063/1.4958724
12. Demni N., Mouayn Z. Analysis of generalized Poisson distributions associated with higher Landau levels. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 18 (4), 1550028 (2015).
13. Zhang H., Liu Y., Li B. Notes on discrete compound Poisson model with applications to risk theory. Insurance Math. Econom. 59, 325–336 (2014).
14. Billingsley P. Convergence of probability measures. New York, Wiley (1999).
15. Feller W. On regular variation and local limit theorems. Proc. V Berkeley Symp. Math. Stats. Prob. 2, part 1, 373–388 (1967).
16. Natanson I.P. Theory of Functions of a Real Variable. Moscow, Nauka Publ. (1974). (In Russian)
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Published
2021-09-26
How to Cite
Alexeev, I. A., & Khartov, A. A. (2021). On convergence and compactness in variation with shift of discrete probability laws. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8(3), 385–393. https://doi.org/10.21638/spbu01.2021.301
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
