Hostname: page-component-784d4fb959-mvngg Total loading time: 0 Render date: 2025-07-16T22:17:57.026Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE CUMULATIVE DISTRIBUTION FUNCTION OF THE VARIANCE-GAMMA DISTRIBUTION

Part of: Distribution theory Distribution theory - Probability

Published online by Cambridge University Press:  29 January 2024

ROBERT E. GAUNT Open the ORCID record for ROBERT E. GAUNT [Opens in a new window]
ROBERT E. GAUNT*
Affiliation:
Department of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
e-mail: robert.gaunt@manchester.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We obtain exact formulas for the cumulative distribution function of the variance-gamma distribution, as infinite series involving the modified Bessel function of the second kind and the modified Lommel function of the first kind. From these formulas, we deduce exact formulas for the cumulative distribution function of the product of two correlated zero-mean normal random variables.

Keywords

variance-gamma distributioncumulative distribution functionproduct of correlated normal random variablesmodified Bessel functionmodified Lommel function

MSC classification

Primary: 62E15: Exact distribution theory
Secondary: 60E05: Distributions

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 2 , October 2024 , pp. 389 - 397
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Azmoodeh, E., Eichelsbacher, P. and Thäle, C., ‘Optimal variance-gamma approximation on the second Wiener chaos’, J. Funct. Anal. 282 (2022), Article no. 109450.10.1016/j.jfa.2022.109450CrossRefGoogle Scholar
Craig, C. C., ‘On the frequency function of $xy$ ’, Ann. Math. Stat. 7 (1936), 1–15.10.1214/aoms/1177732541CrossRefGoogle Scholar
Fischer, A., Gaunt, R. E. and Sarantsev, A., ‘The variance-gamma distribution: A review’, Preprint, 2023, arXiv:2303.05615.Google Scholar
Gaunt, R. E., R. Rates of Convergence of Variance-Gamma Approximations via Stein’s Method. DPhil thesis, University of Oxford, 2013.10.1214/EJP.v19-3020CrossRefGoogle Scholar
Gaunt, R. E., ‘Variance-gamma approximation via Stein’s method’, Electron. J. Probab. 19(38) (2014), 1–33.10.1214/EJP.v19-3020CrossRefGoogle Scholar
Gaunt, R. E., ‘A note on the distribution of the product of zero mean correlated normal random variables’, Stat. Neerl. 73 (2019), 176–179.10.1111/stan.12152CrossRefGoogle Scholar
Gaunt, R. E., ‘Bounds for modified Lommel functions of the first kind and their ratios’, J. Math. Anal. Appl. 486 (2020), Article no. 123893.10.1016/j.jmaa.2020.123893CrossRefGoogle Scholar
Gaunt, R. E., ‘The basic distributional theory for the product of zero mean correlated normal random variables’, Stat. Neerl. 76 (2022), 450–470.10.1111/stan.12267CrossRefGoogle Scholar
Holm, H. and Alouini, M.-S., ‘Sum and difference of two squared correlated Nakagami variates with the McKay distribution’, IEEE Trans. Comput. 52 (2004), 1367–1376.Google Scholar
Jankov Maširević, D. and Pogány, T. K., ‘On new formulae for cumulative distribution function for McKay Bessel distribution’, Commun. Stat. Theory 50 (2021), 143–160.10.1080/03610926.2019.1632898CrossRefGoogle Scholar
Kotz, S., Kozubowski, T. J. and Podgórski, K., The Laplace Distribution and Generalizations: A Revisit with New Applications (Springer, Berlin, 2001).10.1007/978-1-4612-0173-1CrossRefGoogle Scholar
Madan, D. B., Carr, P. and Chang, E. C., ‘The variance gamma process and option pricing’, Eur. Finance Rev. 2 (1998), 74–105.10.1023/A:1009703431535CrossRefGoogle Scholar
Madan, D. B. and Seneta, E., ‘The variance gamma (V.G.) model for share market returns’, J. Bus. 63 (1990), 511–524.10.1086/296519CrossRefGoogle Scholar
McKay, A. T., ‘A Bessel function distribution’, Biometrika 24 (1932), 39–44.10.1093/biomet/24.1-2.39CrossRefGoogle Scholar
Nadarajah, S. and Pogány, T. K., ‘On the distribution of the product of correlated normal random variables’, C. R. Acad. Sci. Paris, Ser. I 354 (2016), 201–204.10.1016/j.crma.2015.10.019CrossRefGoogle Scholar
Nadarajah, S., Srivastava, H. M. and Gupta, A. K., ‘Skewed Bessel function distributions with application to rainfall data’, Statistics 41 (2007), 333–344.10.1080/02331880701270606CrossRefGoogle Scholar
Olver, F. W. J., Lozier, D. W., Boisvert, R. F. and Clark, C. W., NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010).Google Scholar
Pearson, K., Jefferey, G. B. and Elderton, E. M., ‘On the distribution of the first product moment-coefficient, in samples drawn from an indefinitely large normal population’, Biometrika 21(1929), 164–193.10.1093/biomet/21.1-4.164CrossRefGoogle Scholar
Rollinger, C. N., ‘Lommel functions with imaginary argument’, Q. Appl. Math. 21 (1964), 343–349.10.1090/qam/153883CrossRefGoogle Scholar