Exponential and related probability distributions on symmetric matrices. (English) Zbl 1490.60038

Summary: Pursuing the study initiated in [A. Hassairi and A. Roula, Stat. Probab. Lett. 145, 37–42 (2019; Zbl 1414.62174)], we show in the present paper that the reliability function of a probability distribution on the cone \(\varOmega\) of positive definite symmetric matrices characterizes the distribution without any invariance condition. We also show that the characterization of the exponential probability distribution on \(\varOmega\) by a memoryless property holds without assuming an invariance condition. We then study the connection between the exponential distribution on \(\varOmega\) and the uniform distribution on a bounded interval of \(\varOmega\). A notion of matrix Pareto distribution is introduced, and it is shown that this distribution possesses the long tail property.


60E05 Probability distributions: general theory
60B11 Probability theory on linear topological spaces
62E15 Exact distribution theory in statistics


Zbl 1414.62174
Full Text: DOI


[1] Andrews, G. E.; Askey, R.; Roy, R., Special Functions (1999), Cambridge University Press · Zbl 0920.33001
[2] Daher, R.; El Ouadih, S., Characterization of \(( \eta , \gamma , k , 2 )\)-dini-Lipschitz functions in terms if their Helgason Fourier transform, Matematichki Vesnik, 68, 4, 267-276 (2016) · Zbl 1474.42020
[3] Faraut, J.; Korànyi, A., Analysis on Symmetric Cones (1994), Oxford Univ. Press · Zbl 0841.43002
[4] Gupta, A. K.; Nagar, D. K., Matrix Variate Distribution (2000), Chapman Hall/CRC: Chapman Hall/CRC Boca Raton · Zbl 0981.62043
[5] Hassairi, A., Riesz Probability Distributions (2021), De Gruyter · Zbl 1496.60001
[6] Hassairi, A.; Lajmi, S.; Zine, R., A characterization of the Riesz probability distribution, J. Theor. Probab., 21, 773-790 (2008) · Zbl 1153.60008
[7] Hassairi, A.; Roula, A., Exponential probability distribution on symmetric matrices, Statist. Probab. Lett., 145, 37-42 (2019) · Zbl 1414.62174
[8] Janos, G.; Samuel, K., Characterizations of Probability Distributions (1978), Temple University Press
[9] Kennedy, K., Exponential Distribution: Its Constructions, Characterizations and Relates Distributions (2009), University of Nairobi., Thesis
[10] Nagar, D. K.; Mosquera-Benitez, J. C., Properties of matrix variate hypergeometric function distribution, Appl. Math. Sci., 11, 677-692 (2017)
[11] Parasar, M.; Swagato, R.; Rudra, S.; Alladi, S., The Helgason-Fourier transform for symmetric spaces II, J. Lie Theory, 14, 227-242 (2004) · Zbl 1047.43013
[12] Reem, D., Remarks on the Cauchy functional equation and variations of it, Aequationes Math., 91, 2, 237-264 (2017) · Zbl 1368.39017
[13] Terras, A., Harmonic Analysis on Symmetric Spaces and Applications II (1982), University of California at San Diego. Press
[14] Tropp, J. A., User-friendly tail bounds for sums of random matrices, Found. Comput. Math., 12, 389-434 (2012) · Zbl 1259.60008
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.