## Bernoulli

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

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

#### 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
Revised: March 2015
First available in Project Euclid: 3 May 2016

https://projecteuclid.org/euclid.bj/1462297683

Digital Object Identifier
doi:10.3150/15-BEJ731

Mathematical Reviews number (MathSciNet)
MR3498031

Zentralblatt MATH identifier
1352.60065

#### 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

#### 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.