Electronic Communications in Probability

$L_1$-distance for additive processes with time-homogeneous Lévy measures

Abstract

We give an explicit bound for the $L_1$-distance between two additive processes of local characteristics $(f_j(\cdot),\sigma^2(\cdot),\nu_j)$, $j = 1,2$. The cases $\sigma =0$ and $\sigma(\cdot) > 0$ are both treated. We allow $\nu_1$ and $\nu_2$ to be time-homogeneous Lévy measures, possibly with infinite variation. Some examples of possible applications are discussed.<br /><br />

Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 57, 10 pp.

Dates
Accepted: 21 August 2014
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465316759

Digital Object Identifier
doi:10.1214/ECP.v19-3678

Mathematical Reviews number (MathSciNet)
MR3254736

Zentralblatt MATH identifier
1300.60017

Rights

Citation

Etoré, Pierre; Mariucci, Ester. $L_1$-distance for additive processes with time-homogeneous Lévy measures. Electron. Commun. Probab. 19 (2014), paper no. 57, 10 pp. doi:10.1214/ECP.v19-3678. https://projecteuclid.org/euclid.ecp/1465316759

References

• Brown, Lawrence D.; Low, Mark G. Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 (1996), no. 6, 2384–2398.
• Carter, Andrew V. Deficiency distance between multinomial and multivariate normal experiments. Dedicated to the memory of Lucien Le Cam. Ann. Statist. 30 (2002), no. 3, 708–730.
• Cont, Rama; Tankov, Peter. Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL, 2004. xvi+535 pp. ISBN: 1-5848-8413-4
• Dalalyan, Arnak; Reiss, Markus. Asymptotic statistical equivalence for scalar ergodic diffusions. Probab. Theory Related Fields 134 (2006), no. 2, 248–282.
• Jan Gairing, Michael HÃ¶gele, Tetiana Kosenkova, and Alexei Kulik. Coupling distances between lévy measures and applications to noise sensitivity of sde. 2013.
• Genon-Catalot, Valentine; Larédo, Catherine. Asymptotic equivalence of nonparametric diffusion and Euler scheme experiments. Ann. Statist. 42 (2014), no. 3, 1145–1165.
• Alison L. Gibbs and Francis Edward Su. On choosing and bounding probability metrics. Int. Stat. Rev., 70(3):419–435, 2002.
• Jacod, Jean; Shiryaev, Albert N. Limit theorems for stochastic processes. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 288. Springer-Verlag, Berlin, 2003. xx+661 pp. ISBN: 3-540-43932-3
• Le Cam, Lucien. Asymptotic methods in statistical decision theory. Springer Series in Statistics. Springer-Verlag, New York, 1986. xxvi+742 pp. ISBN: 0-387-96307-3
• Le Cam, Lucien; Yang, Grace Lo. Asymptotics in statistics. Some basic concepts. Second edition. Springer Series in Statistics. Springer-Verlag, New York, 2000. xiv+285 pp. ISBN: 0-387-95036-2
• Mémin, J.; Shiryayev, A. N. Distance de Hellinger-Kakutani des lois correspondant Ã deux processus Ã accroissements indépendants. (French) [Hellinger-Kakutani distance of the laws corresponding to two processes with independent increments] Z. Wahrsch. Verw. Gebiete 70 (1985), no. 1, 67–89.
• Milstein, Grigori; Nussbaum, Michael. Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 (1998), no. 4, 535–543.
• Newman, Charles M. The inner product of path space measures corresponding to random processes with independent increments. Bull. Amer. Math. Soc. 78 1972 268–271.
• Nourdin, Ivan; Peccati, Giovanni. Stein's method on Wiener chaos. Probab. Theory Related Fields 145 (2009), no. 1-2, 75–118.
• Nussbaum, Michael. Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 (1996), no. 6, 2399–2430.
• Giovanni Peccati. The Chen-Stein method for Poisson functionals. arXiv preprint arXiv:1112.5051, 2011.
• Ross, Nathan. Fundamentals of Stein's method. Probab. Surv. 8 (2011), 210–293.
• Sato, Ken-iti. Lévy processes and infinitely divisible distributions. Translated from the 1990 Japanese original. Revised by the author. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge, 1999. xii+486 pp. ISBN: 0-521-55302-4
• Ken-iti Sato. Density transformation in Lévy processes. 2000.