Bernoulli

  • Bernoulli
  • Volume 22, Number 4 (2016), 2325-2371.

Distributional representations and dominance of a Lévy process over its maximal jump processes

Boris Buchmann, Yuguang Fan, and Ross A. Maller

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Distributional identities for a Lévy process $X_{t}$, its quadratic variation process $V_{t}$ and its maximal jump processes, are derived, and used to make “small time” (as $t\downarrow0$) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of $X$. Apart from providing insight into the connections between $X$, $V$, and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study “self-normalised” versions of $X_{t}$, that is, $X_{t}$ after division by $\sup_{0<s\le t}\Delta X_{s}$, or by $\sup_{0<s\le t}\vert \Delta X_{s}\vert $. Thus, we obtain necessary and sufficient conditions for $X_{t}/\sup_{0<s\le t}\Delta X_{s}$ and $X_{t}/\sup_{0<s\le t}\vert \Delta X_{s}\vert $ to converge in probability to 1, or to $\infty$, as $t\downarrow0$, so that $X$ is either comparable to, or dominates, its largest jump. The former situation tends to occur when the singularity at 0 of the Lévy measure of $X$ is fairly mild (its tail is slowly varying at 0), while the latter situation is related to the relative stability or attraction to normality of $X$ at 0 (a steeper singularity at 0). An important component in the analyses is the way the largest positive and negative jumps interact with each other. Analogous “large time” (as $t\to\infty$) versions of the results can also be obtained.

Article information

Source
Bernoulli Volume 22, Number 4 (2016), 2325-2371.

Dates
Received: October 2014
Revised: March 2015
First available in Project Euclid: 3 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.bj/1462297683

Digital Object Identifier
doi:10.3150/15-BEJ731

Mathematical Reviews number (MathSciNet)
MR3498031

Keywords
distributional representation domain of attraction to normality dominance Lévy process maximal jump process relative stability

Citation

Buchmann, Boris; Fan, Yuguang; Maller, Ross A. Distributional representations and dominance of a Lévy process over its maximal jump processes. Bernoulli 22 (2016), no. 4, 2325--2371. doi:10.3150/15-BEJ731. https://projecteuclid.org/euclid.bj/1462297683


Export citation

References

  • [1] Andrew, P. (2008). On the limiting behaviour of Lévy processes at zero. Probab. Theory Related Fields 140 103–127.
  • [2] Arov, D.Z. and Bobrov, A.A. (1960). The extreme members of a sample and their role in the sum of independent variables. Theory Probab. Appl. 5 377–396.
  • [3] Berkes, I. and Horváth, L. (2012). The central limit theorem for sums of trimmed variables with heavy tails. Stochastic Process. Appl. 122 449–465.
  • [4] Berkes, I., Horváth, L. and Schauer, J. (2010). Non-central limit theorems for random selections. Probab. Theory Related Fields 147 449–479.
  • [5] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge: Cambridge Univ. Press.
  • [6] Bertoin, J. (1997). Regularity of the half-line for Lévy processes. Bull. Sci. Math. 121 345–354.
  • [7] Bingham, N.H., Goldie, C.M. and Teugels, J.L. (1987). Regular Variation. Encyclopedia of Mathematics and Its Applications 27. Cambridge: Cambridge Univ. Press.
  • [8] Csörgő, S., Haeusler, E. and Mason, D.M. (1988). A probabilistic approach to the asymptotic distribution of sums of independent, identically distributed random variables. Adv. in Appl. Math. 9 259–333.
  • [9] Csörgő, S. and Simons, G. (2002). A Bibliography of the St. Petersburg Paradox. Analysis and Stochastic Research Group of the Hungarian Academy of Sciences and the University of Szeged.
  • [10] Darling, D.A. (1952). The influence of the maximum term in the addition of independent random variables. Trans. Amer. Math. Soc. 73 95–107.
  • [11] Doney, R.A. (2004). Small-time behaviour of Lévy processes. Electron. J. Probab. 9 209–229.
  • [12] Doney, R.A. (2005). Fluctuation theory for Lévy processes: Ecole d’Eté de Probabilités de Saint-Flour XXXV-2005, Issue 1897.
  • [13] Doney, R.A. and Maller, R.A. (2002). Stability and attraction to normality for Lévy processes at zero and at infinity. J. Theoret. Probab. 15 751–792.
  • [14] Doney, R.A. and Maller, R.A. (2002). Stability of the overshoot for Lévy processes. Ann. Probab. 30 188–212.
  • [15] Fan, Y. (2015). A study in lightly trimmed Lévy processes. PhD thesis, the Australian National Univ.
  • [16] Feller, W. (1968/1969). An extension of the law of the iterated logarithm to variables without variance. J. Math. Mech. 18 343–355.
  • [17] Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. New York: Wiley.
  • [18] Fukker, G., Györfi, L. and Kevei, P. (2015). Asymptotic behaviour of the St. Petersburg sum conditioned on its maximum. Bernoulli. To appear.
  • [19] Griffin, P.S. and Maller, R.A. (2011). Stability of the exit time for Lévy processes. Adv. in Appl. Probab. 43 712–734.
  • [20] Griffin, P.S. and Maller, R.A. (2013). Small and large time stability of the time taken for a Lévy process to cross curved boundaries. Ann. Inst. Henri Poincaré Probab. Stat. 49 208–235.
  • [21] Griffin, P.S. and Pruitt, W.E. (1989). Asymptotic normality and subsequential limits of trimmed sums. Ann. Probab. 17 1186–1219.
  • [22] Gut, A. and Martin-Löf, A. (2014). A maxtrimmed St. Petersburg game. J. Theoret. Probab. To appear.
  • [23] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. New York: Springer.
  • [24] Kesten, H. and Maller, R.A. (1992). Ratios of trimmed sums and order statistics. Ann. Probab. 20 1805–1842.
  • [25] Kesten, H. and Maller, R.A. (1994). Infinite limits and infinite limit points of random walks and trimmed sums. Ann. Probab. 22 1473–1513.
  • [26] Kesten, H. and Maller, R.A. (1995). The effect of trimming on the strong law of large numbers. Proc. Lond. Math. Soc. (3) 71 441–480.
  • [27] Kevei, P. and Mason, D.M. (2013). Randomly weighted self-normalized Lévy processes. Stochastic Process. Appl. 123 490–522.
  • [28] Khintchine, A. (1939). Sur la croissance locale des processus stochastiques homogènes à accroissements indépendants. Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR] 1939 487–508.
  • [29] Khintchine, Y.A. (1937). Zur Theorie der unbeschränkt teilbaren Verteilungsgesetze. Mat. Sb. 2 79–119.
  • [30] Klass, M.J. and Wittmann, R. (1993). Which i.i.d. sums are recurrently dominated by their maximal terms? J. Theoret. Probab. 6 195–207.
  • [31] Ladoucette, S.A. and Teugels, J.L. (2007). Asymptotics for ratios with applications to reinsurance. Methodol. Comput. Appl. Probab. 9 225–242.
  • [32] LePage, R. (1980). Multidimensional infinitely divisible variables and processes. I, Technical Rept. 292, Dept. Statistics, Stanford Univ.
  • [33] LePage, R. (1981). Multidimensional infinitely divisible variables and processes. II. In Probability in Banach Spaces, III (Medford, Mass., 1980). Lecture Notes in Math. 860 279–284. Berlin–New York: Springer.
  • [34] LePage, R., Woodroofe, M. and Zinn, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624–632.
  • [35] Madan, D.B. and Seneta, E. (1990). The variance gamma (V.G.) model for share market returns. J. Business 63 511–524.
  • [36] Maller, R. and Mason, D.M. (2008). Convergence in distribution of Lévy processes at small times with self-normalization. Acta Sci. Math. (Szeged) 74 315–347.
  • [37] Maller, R. and Mason, D.M. (2009). Stochastic compactness of Lévy processes. In High Dimensional Probability V: The Luminy Volume. Inst. Math. Stat. Collect. 5 239–257. Beachwood, OH: IMS.
  • [38] Maller, R. and Mason, D.M. (2010). Small-time compactness and convergence behavior of deterministically and self-normalised Lévy processes. Trans. Amer. Math. Soc. 362 2205–2248.
  • [39] Maller, R. and Mason, D.M. (2013). A characterization of small and large time limit laws for self-normalized Lévy processes. In Limit Theorems in Probability, Statistics and Number Theory (P. Eichelsbacher et al., eds.). Springer Proc. Math. Stat. 42 141–169. Heidelberg: Springer.
  • [40] Maller, R.A. (2009). Small-time versions of Strassen’s law for Lévy processes. Proc. Lond. Math. Soc. (3) 98 531–558.
  • [41] Maller, R.A. and Resnick, S.I. (1984). Limiting behaviour of sums and the term of maximum modulus. Proc. Lond. Math. Soc. (3) 49 385–422.
  • [42] Mori, T. (1984). On the limit distributions of lightly trimmed sums. Math. Proc. Cambridge Philos. Soc. 96 507–516.
  • [43] Pruitt, W.E. (1987). The contribution to the sum of the summand of maximum modulus. Ann. Probab. 15 885–896.
  • [44] Resnick, S.I. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer Series in Operations Research and Financial Engineering. New York: Springer.
  • [45] Resnick, S.I. (2008). Extreme Values, Regular Variation and Point Processes. Springer Series in Operations Research and Financial Engineering. New York: Springer. Reprint of the 1987 original.
  • [46] Rosiński, J. (2001). Series representations of Lévy processes from the perspective of point processes. In Lévy Processes 401–415. Boston, MA: Birkhäuser.
  • [47] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
  • [48] Silvestrov, D.S. and Teugels, J.L. (2004). Limit theorems for mixed max-sum processes with renewal stopping. Ann. Appl. Probab. 14 1838–1868.