no. G7
Probabilités
The variance-gamma ratio distribution
Gaunt, Robert E.1  ; Li, Siqi1
1 Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
Comptes Rendus. Mathématique, Tome 361 (2023) no. G7, pp. 1151-1161

Let X and Y be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio X/Y is derived. Some basic distributional properties are also derived, including identification of parameter regimes under which the density is bounded, asymptotic approximations of tail probabilities, and fractional moments; in particular, we see that the mean is undefined. In the case that X and Y are independent symmetric variance-gamma random variables, an exact formula is also given for the cumulative distribution function of the ratio X/Y.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.495
Classification : 60E05, 62E15
Keywords: Variance-gamma distribution, ratio distribution, product of correlated normal random variables, hypergeometric function, Meijer $G$-function

Gaunt, Robert E. 1 ; Li, Siqi 1

1 Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2023__361_G7_1151_0,
     author = {Gaunt, Robert E. and Li, Siqi},
     title = {The variance-gamma ratio distribution},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1151--1161},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     number = {G7},
     doi = {10.5802/crmath.495},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.495/}
}
TY  - JOUR
AU  - Gaunt, Robert E.
AU  - Li, Siqi
TI  - The variance-gamma ratio distribution
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1151
EP  - 1161
VL  - 361
IS  - G7
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.495/
DO  - 10.5802/crmath.495
LA  - en
ID  - CRMATH_2023__361_G7_1151_0
ER  - 
%0 Journal Article
%A Gaunt, Robert E.
%A Li, Siqi
%T The variance-gamma ratio distribution
%J Comptes Rendus. Mathématique
%D 2023
%P 1151-1161
%V 361
%N G7
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.495/
%R 10.5802/crmath.495
%G en
%F CRMATH_2023__361_G7_1151_0
Gaunt, Robert E.; Li, Siqi. The variance-gamma ratio distribution. Comptes Rendus. Mathématique, Tome 361 (2023) no. G7, pp. 1151-1161. doi: 10.5802/crmath.495

[1] Craig, Cecil C. On the frequency function of xy, Ann. Math. Stat., Volume 7 (1936), pp. 1-15 | Zbl | DOI

[2] Cui, Guolong; Yu, Xianxiang; Iommelli, Salvatore; Kong, Lingjiang Exact distribution for the product of two correlated gaussian random variables, IEEE Sel. Top. Signal Process., Volume 23 (2016), pp. 1662-1666

[3] Gaunt, Robert E. Rates of Convergence of Variance-Gamma Approximations via Stein’s Method, Ph. D. Thesis, University of Oxford (2013)

[4] Gaunt, Robert E. Variance-gamma approximation via Stein’s method, Electron. J. Probab., Volume 19 (2014), 38, 33 pages | MR | Zbl

[5] Gaunt, Robert E. A note on the distribution of the product of zero mean correlated normal random variables, Stat. Neerl., Volume 73 (2019) no. 2, pp. 176-179 | DOI | MR

[6] Gaunt, Robert E. The basic distributional theory for the product of zero mean correlated normal random variables, Stat. Neerl., Volume 76 (2022), pp. 450-470 | DOI | MR

[7] Gaunt, Robert E. Stein factors for variance-gamma approximation in the Wasserstein and Kolmogorov distances, J. Math. Anal. Appl., Volume 514 (2022) no. 1, 126274, 32 pages | MR | Zbl

[8] Gaunt, Robert E. On the moments of the variance-gamma distribution, Stat. Probab. Lett. (2023), 109884, 4 pages | MR | Zbl

[9] Gradshtejn, I. S.; Ryzhik, I. M. Table of Integrals, Series and Products, Academic Press Inc., 2007

[10] Grishchuk, L. P. Statistics of the microwave background anisotropies caused by the squeezed cosmological perturbations, Phys. Rev. D, Volume 53 (1996), pp. 6784-6795 | DOI

[11] Holm, Henrik; Alouini, M.-S. Sum and difference of two squared correlated Nakagami variates with the McKay distribution, IEEE Trans. Commun., Volume 52 (2004) no. 8, pp. 1367-1376 | DOI

[12] Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. Continuous Univariate Distribution. Vol. 1, John Wiley & Sons, 1994

[13] Kotz, Samuel; Kozubowski, Tomasz J.; Podgórski, Krysztof The Laplace distribution and generalizations. A revisit with applications to communications, economics, engineering, and finance, Birkhäuser, 2001

[14] Luke, Yudell L. The special functions and their approximations. Vol. I, Mathematics in Science and Engineering, 53, Academic Press Inc., 1969

[15] Madan, Dilip B.; Carr, Peter P.; Chang, Eric C. The variance gamma process and option pricing, Eur. Finance Rev., Volume 2 (1998) no. 1, pp. 74-105 | Zbl

[16] Madan, Dilip B.; Seneta, Eugene The Variance Gamma (V.G.) Model for Share Market Returns, J. Bus., Volume 63 (1990) no. 4, pp. 511-524 | DOI

[17] McKay, A. T. A Bessel function distribution, Biometrika, Volume 24 (1932), pp. 39-44 | DOI | Zbl

[18] Nadarajah, Saralees Exact distribution of the product of N Student’s t RVs, Methodol. Comput. Appl. Probab., Volume 14 (2012) no. 4, pp. 997-1009 | DOI | MR | Zbl

[19] Nadarajah, Saralees; Kotz, Samuel The Bessel ratio distribution, C. R. Acad. Sci. Paris, Volume 343 (2006) no. 8, pp. 531-534 | DOI | MR | Numdam | Zbl

[20] Nadarajah, Saralees; Pogány, Tibor K. On the distribution of the product of correlated normal random variables, C. R. Acad. Sci. Paris, Volume 354 (2016) no. 2, pp. 201-204 | DOI | MR | Numdam | Zbl

[21] NIST Handbook of Mathematical Functions (Olver, Frank W. J.; Lozier, Daniel W.; Boisvert, Ronald F.; Clark, Charles W., eds.), Cambridge University Press, 2010 | Zbl

[22] Seneta, Eugene Fitting the variance-gamma model to financial data, J. Appl. Probab., Volume 41A (2004), pp. 177-187 | DOI | MR | Zbl

Cité par Sources :