Variance-Gamma approximation via Stein's method

Robert Edward Gaunt (University of Oxford)


Variance-Gamma distributions are widely used in financial modelling and contain as special cases the normal, Gamma and Laplace distributions. In this paper we extend Stein's method to this class of distributions. In particular, we obtain a Stein equation and smoothness estimates for its solution. This Stein equation has the attractive property of reducing to the known normal and Gamma Stein equations for certain parameter values. We apply these results and local couplings to bound the distance between sums of the form $\sum_{i,j,k=1}^{m,n,r}X_{ik}Y_{jk}$, where the $X_{ik}$ and $Y_{jk}$ are independent and identically distributed random variables with zero mean, by their limiting Variance-Gamma distribution. Through the use of novel symmetry arguments, we obtain a bound on the distance that is of order $m^{-1}+n^{-1}$ for smooth test functions. We end with a simple application to binary sequence comparison.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-33

Publication Date: March 29, 2014

DOI: 10.1214/EJP.v19-3020


  • Arfken, George. Mathematical methods for physicists. Academic Press, New York-London 1966 xvi+654 pp. MR0205512
  • Barbour, A. D. Stein's method and Poisson process convergence. A celebration of applied probability. J. Appl. Probab. 1988, Special Vol. 25A, 175--184. MR0974580
  • Barbour, A. D. Stein's method for diffusion approximations. Probab. Theory Related Fields 84 (1990), no. 3, 297--322. MR1035659
  • Barndorff-Nielsen, O.; Halgreen, Christian. Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 38 (1977), no. 4, 309--311. MR0436260
  • Barndorff-Nielsen, O.; Kent, J.; Sorensen, M. Normal variance-mean mixtures and $z$ distributions. Internat. Statist. Rev. 50 (1982), no. 2, 145--159. MR0678296
  • Berry, Andrew C. The accuracy of the Gaussian approximation to the sum of independent variates. Trans. Amer. Math. Soc. 49, (1941). 122--136. MR0003498
  • BIBBY, B. M., and SORENSEN, M. Hyperbolic Processes in Finance. In S. Rachev (ed.), Handbook of Heavy Tailed Distributions in Finance (2003), pp. 211--248. Amsterdam: Elsevier Science.
  • BLAISDELL, B. A measure of the similarity of sets of sequences not requiring sequence alignment. Proc. Natl. Acad. Sci. USA 83 (1986), pp. 5155--5159.
  • Chatterjee, Sourav; Fulman, Jason; Rollin, Adrian. Exponential approximation by Stein's method and spectral graph theory. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011), 197--223. MR2802856
  • Chen, Louis H. Y. Poisson approximation for dependent trials. Ann. Probability 3 (1975), no. 3, 534--545. MR0428387
  • Chen, Louis H. Y.; Goldstein, Larry; Shao, Qi-Man. Normal approximation by Stein's method. Probability and its Applications (New York). Springer, Heidelberg, 2011. xii+405 pp. ISBN: 978-3-642-15006-7 MR2732624
  • COLLINS, P. J. Differential and Integral Equations. Oxford University Press, 2006.
  • DÖBLER, C. Distributional transformations without orthogonality relations. arXiv:1312.6093, 2013.
  • Durrett, Richard. Stochastic calculus. A practical introduction. Probability and Stochastics Series. CRC Press, Boca Raton, FL, 1996. x+341 pp. ISBN: 0-8493-8071-5 MR1398879
  • Eberlein, Ernst; v. Hammerstein, Ernst August. Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. Seminar on Stochastic Analysis, Random Fields and Applications IV, 221--264, Progr. Probab., 58, Birkhauser, Basel, 2004. MR2096291
  • Esseen, Carl-Gustav. Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math. 77, (1945). 1--125. MR0014626
  • Finlay, Richard; Seneta, Eugene. Option pricing with VG-like models. Int. J. Theor. Appl. Finance 11 (2008), no. 8, 943--955. MR2492088
  • GAUNT, R. E. phRates of Convergence of Variance-Gamma Approximations via Stein's Method. DPhil thesis, University of Oxford, 2013.
  • GAUNT, R. E. Uniform bounds for expressions involving modified Bessel functions. Preprint 2013.
  • GAUNT, R. E. On Stein's Method for products of normal, gamma and beta random variables and a generalisation of the zero bias coupling. Preprint 2014.
  • Goldstein, Larry; Reinert, Gesine. Stein's method and the zero bias transformation with application to simple random sampling. Ann. Appl. Probab. 7 (1997), no. 4, 935--952. MR1484792
  • Goldstein, Larry; Rinott, Yosef. Multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Probab. 33 (1996), no. 1, 1--17. MR1371949
  • HOLM, H. and ALOUINI, M--S. Sum and Difference of two squared correlated Nakagami variates with the McKay distribution. phIEEE Transactions on Communications. 52 (2004), pp 1367-1376.
  • LEY, C. and SWAN, Y. A unified approach to Stein characterizations. arXiv:1105.4925, 2011.
  • Linetsky, Vadim. The spectral representation of Bessel processes with constant drift: applications in queueing and finance. J. Appl. Probab. 41 (2004), no. 2, 327--344. MR2052575
  • Lippert, Ross A.; Huang, Haiyan; Waterman, Michael S. Distributional regimes for the number of $k$-word matches between two random sequences. Proc. Natl. Acad. Sci. USA 99 (2002), no. 22, 13980--13989 (electronic). MR1944413
  • Luk, Ho Ming. Stein's method for the Gamma distribution and related statistical applications. Thesis (Ph.D.)–University of Southern California. ProQuest LLC, Ann Arbor, MI, 1994. 74 pp. MR2693204
  • MADAN, D. B. and SENETA, E. The Variance Gamma (V.G.) Model for Share Market Returns. Journal of Business 63 (1990), pp. 511--524.
  • Nourdin, Ivan; Peccati, Giovanni. Stein's method on Wiener chaos. Probab. Theory Related Fields 145 (2009), no. 1-2, 75--118. MR2520122
  • NIST handbook of mathematical functions. Edited by Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark. With 1 CD-ROM (Windows, Macintosh and UNIX). U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. xvi+951 pp. ISBN: 978-0-521-14063-8 MR2723248
  • Pekez, Erol A.; Rollin, Adrian. New rates for exponential approximation and the theorems of Rényi and Yaglom. Ann. Probab. 39 (2011), no. 2, 587--608. MR2789507
  • Pekez, Erol A.; Rollin, Adrian; Ross, Nathan. Degree asymptotics with rates for preferential attachment random graphs. Ann. Appl. Probab. 23 (2013), no. 3, 1188--1218. MR3076682
  • PICKETT, A. phRates of Convergence of χ^2 Approximations via Stein's Method. DPhil thesis, University of Oxford, 2004.
  • PIKE, J. and REN, H. Stein's method and the Laplace distribution. arXiv:1210.5775, 2012.
  • RAIC, M. Normal approximation by Stein's method. In: Proceedings of the 7th Young Statisticians Meeting (2003), pp. 71--97.
  • Reinert, Gesine. Three general approaches to Stein's method. An introduction to Stein's method, 183--221, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 4, Singapore Univ. Press, Singapore, 2005. MR2235451
  • Lothaire, M. Applied combinatorics on words. A collective work by Jean Berstel, Dominique Perrin, Maxime Crochemore, Eric Laporte, Mehryar Mohri, Nadia Pisanti, Marie-France Sagot, Gesine Reinert, Sophie Schbath, Michael Waterman, Philippe Jacquet, Wojciech Szpankowski, Dominique Poulalhon, Gilles Schaeffer, Roman Kolpakov, Gregory Koucherov, Jean-Paul Allouche and Valérie Berthé. With a preface by Berstel and Perrin. Encyclopedia of Mathematics and its Applications, 105. Cambridge University Press, Cambridge, 2005. xvi+610 pp. ISBN: 978-0-521-84802-2; 0-521-84802-4 MR2165687
  • Reinert, Gesine; Rollin, Adrian. Multivariate normal approximation with Stein's method of exchangeable pairs under a general linearity condition. Ann. Probab. 37 (2009), no. 6, 2150--2173. MR2573554
  • Reinert, Gesine; Chew, David; Sun, Fengzhu; Waterman, Michael S. Alignment-free sequence comparison. I. Statistics and power. J. Comput. Biol. 16 (2009), no. 12, 1615--1634. MR2578699
  • Scott, David J.; Wurtz, Diethelm; Dong, Christine; Tran, Thanh Tam. Moments of the generalized hyperbolic distribution. Comput. Statist. 26 (2011), no. 3, 459--476. MR2833142
  • Stein, Charles. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability theory, pp. 583--602. Univ. California Press, Berkeley, Calif., 1972. MR0402873
  • Stein, Charles. Approximate computation of expectations. Institute of Mathematical Statistics Lecture Notes—Monograph Series, 7. Institute of Mathematical Statistics, Hayward, CA, 1986. iv+164 pp. ISBN: 0-940600-08-0 MR0882007
  • Stein, Charles; Diaconis, Persi; Holmes, Susan; Reinert, Gesine. Use of exchangeable pairs in the analysis of simulations. Stein's method: expository lectures and applications, 1--26, IMS Lecture Notes Monogr. Ser., 46, Inst. Math. Statist., Beachwood, OH, 2004. MR2118600
  • WINKELBAUER, A. Moments and absolute moments of the normal distribution, arXiv:1209.4340, 2012.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.