# Convex rearrangements of Lévy processes

Youri Davydov; Emmanuel Thilly

ESAIM: Probability and Statistics (2007)

- Volume: 11, page 161-172
- ISSN: 1292-8100

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topDavydov, Youri, and Thilly, Emmanuel. "Convex rearrangements of Lévy processes." ESAIM: Probability and Statistics 11 (2007): 161-172. <http://eudml.org/doc/250127>.

@article{Davydov2007,

abstract = {
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 + with exponent α ∈ (1,2).},

author = {Davydov, Youri, Thilly, Emmanuel},

journal = {ESAIM: Probability and Statistics},

keywords = {Convex rearrangements; Lévy processes; strong
laws; Lorenz curve; regularly varying functions.; convex rearrangements; strong laws; regularly varying functions},

language = {eng},

month = {3},

pages = {161-172},

publisher = {EDP Sciences},

title = {Convex rearrangements of Lévy processes},

url = {http://eudml.org/doc/250127},

volume = {11},

year = {2007},

}

TY - JOUR

AU - Davydov, Youri

AU - Thilly, Emmanuel

TI - Convex rearrangements of Lévy processes

JO - ESAIM: Probability and Statistics

DA - 2007/3//

PB - EDP Sciences

VL - 11

SP - 161

EP - 172

AB -
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 + with exponent α ∈ (1,2).

LA - eng

KW - Convex rearrangements; Lévy processes; strong
laws; Lorenz curve; regularly varying functions.; convex rearrangements; strong laws; regularly varying functions

UR - http://eudml.org/doc/250127

ER -

## References

top- J-M. Azaïs and M. Wschebor, Almost sure oscillation of certain random processes. Bernoulli2 (1996) 257–270.
- J. Bertoin, Lévy processes. Cambridge University Press (1998).
- N.J. Bingham, C.M. Goldie and J.L. Teugels, Regular variation. Cambridge University Press (1987).
- 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 Inference74 (1998) 65–91.
- 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.
- M. Csörgö and R. Zitikis, On the rate of strong consistency of Lorenz curves. Statist. Probab. Lett.34 (1997) 113–121.
- M. Csörgö and R. Zitikis, Strassen's LIL for the Lorenz curve. J. Multivariate Anal.59 (1996) 1–12.
- Y. Davydov, Convex rearrangements of stable processes. J. Math. Sci.92 (1998) 4010–4016.
- Y. Davydov and V. Egorov, Functional limit theorems for induced order statistics. Math. Methods Stat.9 (2000) 297–313.
- Y. Davydov, D. Khoshnevisan, Zh. Shi and R. Zitikis, Convex Rearrangements, Generalized Lorenz Curves, and Correlated Gaussian Data. J. Statistical Planning and Inference137 (2006) 915–934.
- Y. Davydov and E. Thilly, Convex rearrangements of Gaussian processes. Theory Probab. Appl.47 (2002) 219–235.
- 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.
- Y. Davydov and A.M. Vershik, Réarrangements convexes des marches aléatoires. Ann. Inst. Henri Poincaré, Probab. Stat.34 (1998) 73–95.
- Y. Davydov and R. Zitikis, Generalized Lorenz curves and convexifications of stochastic processes. J. Appl. Probab.40 (2003) 906–925.
- Y. Davydov and R. Zitikis, Convex rearrangements of random elements, in Asymptotic Methods in Stochastics. American Mathematical Society, Providence, RI (2004) 141–171.
- 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.
- W. Feller, An introduction to probability theory and its applications, Vol. I and II. John Wiley and Sons Ed. (1968).
- I.I. Gihman and A.V. Skorohod, Introduction to the theory of random processes. W. B. Saunders Co., Philadelphia, PA (1969).
- W. Linde, Probability in Banach Spaces – Stable and Infinitely Divisible Distributions. Wiley, Chichester (1986).
- A. Philippe and E. Thilly, Identification of locally self-similar Gaussian process by using convex rearrangements. Methodol. Comput. Appl. Probab.4 (2002) 195–209.
- B. Ramachandran, On characteristic functions and moments. Sankhya31 Series A (1969) 1–12.
- M. Wschebor, Almost sure weak convergence of the increments of Lévy processes. Stochastic Proc. App.55 (1995) 253–270.
- M. Wschebor, Smoothing and occupation measures of stochastic processes. Ann. Fac. Sci. Toulouse, Math15 (2006) 125–156.

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