Convex rearrangements of Lévy processes
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 161-172

In this paper we study asymptotic behavior of convex rearrangements of Lévy processes. In particular we obtain Glivenko-Cantelli-type strong limit theorems for the convexifications when the corresponding Lévy measure is regularly varying at 0 + with exponent α∈(1,2).

DOI : 10.1051/ps:2007011
Classification : 60G51, 60G52, 60G17
Keywords: convex rearrangements, Lévy processes, strong laws, Lorenz curve, regularly varying functions 
@article{PS_2007__11__161_0,
     author = {Davydov, Youri and Thilly, Emmanuel},
     title = {Convex rearrangements of {L\'evy} processes},
     journal = {ESAIM: Probability and Statistics},
     pages = {161--172},
     year = {2007},
     publisher = {EDP Sciences},
     volume = {11},
     doi = {10.1051/ps:2007011},
     mrnumber = {2299653},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/ps:2007011/}
}
TY  - JOUR
AU  - Davydov, Youri
AU  - Thilly, Emmanuel
TI  - Convex rearrangements of Lévy processes
JO  - ESAIM: Probability and Statistics
PY  - 2007
SP  - 161
EP  - 172
VL  - 11
PB  - EDP Sciences
UR  - https://www.numdam.org/articles/10.1051/ps:2007011/
DO  - 10.1051/ps:2007011
LA  - en
ID  - PS_2007__11__161_0
ER  - 
%0 Journal Article
%A Davydov, Youri
%A Thilly, Emmanuel
%T Convex rearrangements of Lévy processes
%J ESAIM: Probability and Statistics
%D 2007
%P 161-172
%V 11
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/ps:2007011/
%R 10.1051/ps:2007011
%G en
%F PS_2007__11__161_0
Davydov, Youri; Thilly, Emmanuel. Convex rearrangements of Lévy processes. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 161-172. doi: 10.1051/ps:2007011

[1] J-M. Azaïs and M. Wschebor, Almost sure oscillation of certain random processes. Bernoulli 2 (1996) 257-270. | Zbl

[2] J. Bertoin, Lévy processes. Cambridge University Press (1998). | Zbl | MR

[3] N.J. Bingham, C.M. Goldie and J.L. Teugels, Regular variation. Cambridge University Press (1987). | Zbl | MR

[4] M. Csörgö, J.L. Gastwirth and R. Zitikis, Asymptotic confidence bands for the Lorenz and Bonferroni curves based on the empirical Lorenz curve. J. Statistical Planning and Inference 74 (1998) 65-91. | Zbl

[5] M. Csörgö and R. Zitikis, On confidence bands for the Lorenz and Goldie curves, in Advances in the theory and practice of statistics. Wiley, New York (1997) 261-281. | Zbl

[6] M. Csörgö and R. Zitikis, On the rate of strong consistency of Lorenz curves. Statist. Probab. Lett. 34 (1997) 113-121. | Zbl

[7] M. Csörgö and R. Zitikis, Strassen's LIL for the Lorenz curve. J. Multivariate Anal. 59 (1996) 1-12. | Zbl

[8] Y. Davydov, Convex rearrangements of stable processes. J. Math. Sci. 92 (1998) 4010-4016. | Zbl

[9] Y. Davydov and V. Egorov, Functional limit theorems for induced order statistics. Math. Methods Stat. 9 (2000) 297-313. | Zbl

[10] Y. Davydov, D. Khoshnevisan, Zh. Shi and R. Zitikis, Convex Rearrangements, Generalized Lorenz Curves, and Correlated Gaussian Data. J. Statistical Planning and Inference 137 (2006) 915-934. | Zbl

[11] Y. Davydov and E. Thilly, Convex rearrangements of Gaussian processes. Theory Probab. Appl. 47 (2002) 219-235. | Zbl

[12] Y. Davydov and E. Thilly, Convex rearrangements of smoothed random processes, in Limit theorems in probability and statistics. Fourth Hungarian colloquium on limit theorems in probability and statistics, Balatonlelle, Hungary, June 28-July 2, 1999. Vol I. I. Berkes et al., Eds. Janos Bolyai Mathematical Society, Budapest (2002) 521-552. | Zbl

[13] Y. Davydov and A.M. Vershik, Réarrangements convexes des marches aléatoires. Ann. Inst. Henri Poincaré, Probab. Stat. 34 (1998) 73-95. | Zbl | Numdam

[14] Y. Davydov and R. Zitikis, Generalized Lorenz curves and convexifications of stochastic processes. J. Appl. Probab. 40 (2003) 906-925. | Zbl

[15] Y. Davydov and R. Zitikis, Convex rearrangements of random elements, in Asymptotic Methods in Stochastics. American Mathematical Society, Providence, RI (2004) 141-171. | Zbl

[16] R.A. Doney and R.A. Maller, Stability and attraction to normality for Lévy processes at zero and at infinity. J. Theor. Probab. 15 (2002) 751-792. | Zbl

[17] W. Feller, An introduction to probability theory and its applications, Vol. I and II. John Wiley and Sons Ed. (1968). | Zbl | MR

[18] I.I. Gihman and A.V. Skorohod, Introduction to the theory of random processes. W. B. Saunders Co., Philadelphia, PA (1969). | MR

[19] W. Linde, Probability in Banach Spaces - Stable and Infinitely Divisible Distributions. Wiley, Chichester (1986). | Zbl

[20] A. Philippe and E. Thilly, Identification of locally self-similar Gaussian process by using convex rearrangements. Methodol. Comput. Appl. Probab. 4 (2002) 195-209. | Zbl

[21] B. Ramachandran, On characteristic functions and moments. Sankhya 31 Series A (1969) 1-12. | Zbl

[22] M. Wschebor, Almost sure weak convergence of the increments of Lévy processes. Stochastic Proc. App. 55 (1995) 253-270. | Zbl

[23] M. Wschebor, Smoothing and occupation measures of stochastic processes. Ann. Fac. Sci. Toulouse, Math 15 (2006) 125-156. | Zbl | Numdam

Cité par Sources :