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 : https://doi.org/10.1051/ps:2007011
Classification : 60G51,  60G52,  60G17
Mots clés : 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},
     publisher = {EDP-Sciences},
     volume = {11},
     year = {2007},
     doi = {10.1051/ps:2007011},
     mrnumber = {2299653},
     language = {en},
     url = {http://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
DA  - 2007///
SP  - 161
EP  - 172
VL  - 11
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2007011/
UR  - https://www.ams.org/mathscinet-getitem?mr=2299653
UR  - https://doi.org/10.1051/ps:2007011
DO  - 10.1051/ps:2007011
LA  - en
ID  - PS_2007__11__161_0
ER  - 
Davydov, Youri; Thilly, Emmanuel. Convex rearrangements of Lévy processes. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 161-172. doi : 10.1051/ps:2007011. http://www.numdam.org/articles/10.1051/ps:2007011/

[1] J-M. Azaïs and M. Wschebor, Almost sure oscillation of certain random processes. Bernoulli 2 (1996) 257-270. | Zbl 0885.60018

[2] J. Bertoin, Lévy processes. Cambridge University Press (1998). | MR 1406564 | Zbl 0938.60005

[3] N.J. Bingham, C.M. Goldie and J.L. Teugels, Regular variation. Cambridge University Press (1987). | MR 898871 | Zbl 0617.26001

[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 0953.62045

[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 0887.62053

[6] M. Csörgö and R. Zitikis, On the rate of strong consistency of Lorenz curves. Statist. Probab. Lett. 34 (1997) 113-121. | Zbl 0894.60031

[7] M. Csörgö and R. Zitikis, Strassen's LIL for the Lorenz curve. J. Multivariate Anal. 59 (1996) 1-12. | Zbl 0868.60026

[8] Y. Davydov, Convex rearrangements of stable processes. J. Math. Sci. 92 (1998) 4010-4016. | Zbl 0946.60051

[9] Y. Davydov and V. Egorov, Functional limit theorems for induced order statistics. Math. Methods Stat. 9 (2000) 297-313. | Zbl 1010.60031

[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 1107.60017

[11] Y. Davydov and E. Thilly, Convex rearrangements of Gaussian processes. Theory Probab. Appl. 47 (2002) 219-235. | Zbl 1033.60048

[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 1023.60069

[13] Y. Davydov and A.M. Vershik, Réarrangements convexes des marches aléatoires. Ann. Inst. Henri Poincaré, Probab. Stat. 34 (1998) 73-95. | Numdam | Zbl 0903.60006

[14] Y. Davydov and R. Zitikis, Generalized Lorenz curves and convexifications of stochastic processes. J. Appl. Probab. 40 (2003) 906-925. | Zbl 1054.60044

[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 1080.60055

[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 1015.60043

[17] W. Feller, An introduction to probability theory and its applications, Vol. I and II. John Wiley and Sons Ed. (1968). | MR 228020 | Zbl 0155.23101

[18] I.I. Gihman and A.V. Skorohod, Introduction to the theory of random processes. W. B. Saunders Co., Philadelphia, PA (1969). | MR 247660

[19] W. Linde, Probability in Banach Spaces - Stable and Infinitely Divisible Distributions. Wiley, Chichester (1986). | Zbl 0665.60005

[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 1008.60053

[21] B. Ramachandran, On characteristic functions and moments. Sankhya 31 Series A (1969) 1-12. | Zbl 0176.48404

[22] M. Wschebor, Almost sure weak convergence of the increments of Lévy processes. Stochastic Proc. App. 55 (1995) 253-270. | Zbl 0813.60069

[23] M. Wschebor, Smoothing and occupation measures of stochastic processes. Ann. Fac. Sci. Toulouse, Math 15 (2006) 125-156. | Numdam | Zbl 1121.62072

Cité par Sources :