Comparing Fréchet and positive stable laws

Thomas Simon (Université Lille 1)

Abstract


Let ${\bf L}$ be the unit exponential random variable and ${\bf Z}_\alpha$ the standard positive $\alpha$-stable random variable. We prove that $\{(1-\alpha)\alpha^{\gamma_\alpha} {\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha <1\}$ is decreasing for the optimal stochastic order and that $\{(1-\alpha){\bf Z}_\alpha^{ \gamma_\alpha}, 0< \alpha < 1\}$ is increasing for the convex order, with $\gamma_\alpha = \alpha/(1-\alpha).$ We also show that $\{\Gamma(1+\alpha) {\bf Z}_\alpha^{-\alpha}, 1/2\le \alpha \le 1\}$ is decreasing for the convex order, that ${\bf Z}_\alpha^{ \alpha}\,\prec_{st}\, \Gamma(1-\alpha) {\bf L}$ and that $\Gamma(1+\alpha){\bf Z}_\alpha^{-\alpha} \,\prec_{cx}\,{\bf L}.$ This allows to compare ${\bf Z}_\alpha$ with the two extremal Fréchet distributions corresponding to the behaviour of its density at zero and at infinity. We also discuss the applications of these bounds to the strange behaviour of the median of ${\bf Z}_\alpha$ and ${\bf Z}_\alpha^{-\alpha}$ and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of ${\bf Z}_\alpha$ for $\alpha$ rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.

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

Pages: 1-25

Publication Date: January 28, 2014

DOI: 10.1214/EJP.v19-3058

References

  • Basu, S.; DasGupta, A. The mean, median, and mode of unimodal distributions: a characterization. Teor. Veroyatnost. i Primenen. 41 (1996), no. 2, 336--352; translation in Theory Probab. Appl. 41 (1996), no. 2, 210--223 (1997) MR1445756
  • Bercovici, Hari; Pata, Vittorino. Stable laws and domains of attraction in free probability theory. With an appendix by Philippe Biane. Ann. of Math. (2) 149 (1999), no. 3, 1023--1060. MR1709310
  • Bingham, N. H. Limit theorems for occupation times of Markov processes. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 17 1971 1--22. MR0281255
  • Bingham, N. H. Fluctuation theory in continuous time. Advances in Appl. Probability 7 (1975), no. 4, 705--766. MR0386027
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2 MR0898871
  • Cressie, Noel. A note on the behaviour of the stable distributions for small index $\alpha $. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 (1975/76), no. 1, 61--64. MR0380928
  • Demni, Nizar. Kanter random variable and positive free stable distributions. Electron. Commun. Probab. 16 (2011), 137--149. MR2783335
  • Dharmadhikari, Sudhakar; Joag-Dev, Kumar. Unimodality, convexity, and applications. Probability and Mathematical Statistics. Academic Press, Inc., Boston, MA, 1988. xiv+278 pp. ISBN: 0-12-214690-5 MR0954608
  • A. Erdelyi. Higher transcendental functions. Vol III. McGraw-Hill, 1953.
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp. MR0270403
  • Haubold, H. J.; Mathai, A. M.; Saxena, R. K. Mittag-Leffler functions and their applications. J. Appl. Math. 2011, Art. ID 298628, 51 pp. MR2800586
  • Hatzinikitas, Agapitos; Pachos, Jiannis K. One-dimensional stable probability density functions for rational index $0<\alpha\leq 2$. Ann. Physics 323 (2008), no. 12, 3000--3019. MR2467080
  • Khoffman-Iënsen, I. Stable densities. (Russian) Teor. Veroyatnost. i Primenen. 38 (1993), no. 2, 470--476; translation in Theory Probab. Appl. 38 (1993), no. 2, 350--355 MR1317993
  • Kanter, Marek. Stable densities under change of scale and total variation inequalities. Ann. Probability 3 (1975), no. 4, 697--707. MR0436265
  • F. Mainardi. On some properties of the Mittag-Leffler function Ea(-t^a). To appear in Discrete and Continuous Dynamical Systems, Series B. arXiv:1305.0161.
  • G. Mittag-Leffler. Sur la nouvelle fonction Ea(x). Comptes Rendus Acad. Sci. Paris 137, 554-558, 1903.
  • W. Ml otkowski and K. A. Penson. Probability distributions with binomial moments. arXiv:1309.0595.
  • Nagaev, A. V.; Shcolnick, S. M. Properties of mode of spectral positive stable distributions. Stability problems for stochastic models (Varna, 1985), 69--78, Lecture Notes in Math., 1233, Springer, Berlin, 1987. MR0886282
  • Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4 MR1739520
  • Sato, Ken-iti; Yamazato, Makoto. On distribution functions of class $L$. Z. Wahrsch. Verw. Gebiete 43 (1978), no. 4, 273--308. MR0494405
  • Shaked, Moshe; Shanthikumar, J. George. Stochastic orders and their applications. Probability and Mathematical Statistics. Academic Press, Inc., Boston, MA, 1994. xvi+545 pp. ISBN: 0-12-638160-7 MR1278322
  • Simon, Thomas. Fonctions de Mittag-Leffler et processus de Lévy stables sans sauts négatifs. (French) [Mittag-Leffler functions and stable Levy processes without negative jumps] Expo. Math. 28 (2010), no. 3, 290--298. MR2671005
  • Simon, Thomas. Multiplicative strong unimodality for positive stable laws. Proc. Amer. Math. Soc. 139 (2011), no. 7, 2587--2595. MR2784828
  • Simon, Thomas. A multiplicative short proof for the unimodality of stable densities. Electron. Commun. Probab. 16 (2011), 623--629. MR2846655
  • Simon, Thomas. On the unimodality of power transformations of positive stable densities. Math. Nachr. 285 (2012), no. 4, 497--506. MR2899640
  • Stone, Charles. The set of zeros of a semistable process. Illinois J. Math. 7 1963 631--637. MR0158439
  • Williams, E. J. Some representations of stable random variables as products. Biometrika 64 (1977), no. 1, 167--169. MR0448484
  • Zolotarev, V. M. One-dimensional stable distributions. Translated from the Russian by H. H. McFaden. Translation edited by Ben Silver. Translations of Mathematical Monographs, 65. American Mathematical Society, Providence, RI, 1986. x+284 pp. ISBN: 0-8218-4519-5 MR0854867
  • Zolotarev, V. M. On the representation of the densities of stable laws by special functions. (Russian) Teor. Veroyatnost. i Primenen. 39 (1994), no. 2, 429--437; translation in Theory Probab. Appl. 39 (1994), no. 2, 354--362 (1995) MR1404693


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