Comptes Rendus
Probabilités
Exponential inequalities for the supremum of some counting processes and their square martingales
Ronan Le Guével1 
1 Université Rennes 2, France.
Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 969-982.

Nous établissons ici des inégalités exponentielles pour le supremum de martingales et de martingales carrées issues de processus de comptage, ainsi que pour le processus d’oscillation de ces processus. Ces inégalités, qui jouent un rôle essentiel dans le contrôle d’erreur de certaines procédures statistiques, s’appliquent à des processus de comptage non-explosifs généraux, comme les processus de Poisson, de Hawkes ou encore les processsus de Cox. Quelques applications aux U-statistiques sont aussi abordées dans cet article.

We establish exponential inequalities for the supremum of martingales and square martingales obtained from counting processes, as well as for the oscillation modulus of these processes. Our inequalities, that play a decisive role in the control of errors in statistical procedures, apply to general non-explosive counting processes including Poisson, Hawkes and Cox models. Some applications for U-statistics are discussed.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.206
Classification : 60G55
Ronan Le Guével 1

1 Université Rennes 2, France.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2021__359_8_969_0,
     author = {Ronan Le Gu\'evel},
     title = {Exponential inequalities for the supremum of some counting processes and their square martingales},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {969--982},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {8},
     year = {2021},
     doi = {10.5802/crmath.206},
     language = {en},
}
TY  - JOUR
AU  - Ronan Le Guével
TI  - Exponential inequalities for the supremum of some counting processes and their square martingales
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 969
EP  - 982
VL  - 359
IS  - 8
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.206
LA  - en
ID  - CRMATH_2021__359_8_969_0
ER  - 
%0 Journal Article
%A Ronan Le Guével
%T Exponential inequalities for the supremum of some counting processes and their square martingales
%J Comptes Rendus. Mathématique
%D 2021
%P 969-982
%V 359
%N 8
%I Académie des sciences, Paris
%R 10.5802/crmath.206
%G en
%F CRMATH_2021__359_8_969_0
Ronan Le Guével. Exponential inequalities for the supremum of some counting processes and their square martingales. Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 969-982. doi : 10.5802/crmath.206. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.206/

[1] Miguel A. Arcones; Evarist Giné Limit theorems for U-processes, Ann. Probab., Volume 21 (1993) no. 3, pp. 1494-1542 | MR | Zbl

[2] Richard F. Bass The measurability of hitting times, Electron. Commun. Probab., Volume 15 (2010), pp. 99-105 | DOI | MR | Zbl

[3] Richard F. Bass Stochastic processes, Cambridge Series in Statistical and Probabilistic Mathematics, 33, Cambridge University Press, 2011 | Zbl

[4] Bernard Bercu; Bernard Delyon; Emmanuel Rio Concentration Inequalities for Sums and Martingales, SpringerBriefs in Mathematics, Springer, 2015 | Zbl

[5] Pierre Brémaud Point processes and queues: martingale dynamics, Springer Series in Statistics, Springer, 1981 | Zbl

[6] Pierre Brémaud; Laurent Massoulié Stability of nonlinear Hawkes processes, Ann. Probab., Volume 24 (1996) no. 3, pp. 1563-1588 | MR | Zbl

[7] Jean Bretagnolle A new large deviation inequality for U-statistics of order 2, ESAIM, Probab. Stat., Volume 3 (1999), pp. 151-162 | DOI | Numdam | MR | Zbl

[8] Magalie Fromont; Béatrice Laurent Adaptive goodness-of-fit tests in a density model, Ann. Stat., Volume 34 (2006) no. 2, pp. 680-720 | MR | Zbl

[9] Magalie Fromont; Béatrice Laurent; Patricia Reynaud-Bouret Adaptive test of homogeneity for a Poisson process, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 47 (2011) no. 1, pp. 176-213 | Numdam | MR | Zbl

[10] Evarist Giné; Rafał Latała; Joel Zinn Exponential and Moment Inequalities for U-Statistics, High Dimensional Probability II. 2nd international conference, Univ. of Washington, DC, USA, August 1-6, 1999 (Progress in Probability), Volume 47, Birkhäuser, 2000, pp. 13-88 | MR | Zbl

[11] Evarist Giné; Joel Zinn On Hoffmann–Jorgensen’s Inequality for U-Processes, Probability in Banach Spaces, 8. Proceedings of the Eighth International Conference, held at Bowdoin College in summer of 1991 (Progress in Probability), Volume 30, Birkhäuser, 1992, pp. 80-91 | Zbl

[12] David L. Hanson; F. T. Wright A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Stat., Volume 42 (1971) no. 3, pp. 1079-1083 | DOI | MR | Zbl

[13] Wassily Hoeffding Probability Inequalities for Sums of Bounded Random Variables, J. Am. Stat. Assoc., Volume 58 (1963) no. 301, pp. 13-30 | DOI | MR | Zbl

[14] Christian Houdré; Patricia Reynaud-Bouret Exponential Inequalities for U-Statistics of Order Two with Constants, Stochastic Inequalities and Applications. Selected papers presented at the Euroconference on “Stochastic inequalities and their applications”, Barcelona, June 18–22, 2002 (Evariste ad others Giné, ed.) (Progress in Probability), Volume 56, Birkhäuser, 2003, pp. 55-69 | Zbl

[15] Olav Kallenberg Foundations of Modern Probability, Probability and Its Applications, Springer, 1997 | Zbl

[16] Michael J. Klass; Krzysztof Nowicki Order of magnitude bounds for expectations of Δ 2 functions of nonnegative random bilinear forms and generalized U-statistics, Ann. Probab., Volume 25 (1997), pp. 1471-1501 | MR | Zbl

[17] Béatrice Laurent Adaptive estimation of a quadratic functional of a density by model selection, ESAIM, Probab. Stat., Volume 9 (2005), pp. 1-18 | DOI | Numdam | MR | Zbl

[18] Michel Ledoux On Talagrand’s deviation inequalities for product measures, ESAIM, Probab. Stat., Volume 1 (1997), pp. 63-87 | DOI | Numdam | MR | Zbl

[19] Pascal Massart About the constants in Talagrand’s concentration inequalities for empirical processes, Ann. Probab., Volume 28 (2000) no. 2, pp. 863-884 | MR | Zbl

[20] Pascal Massart Concentration Inequalities and Model Selection: École d’Été de Probabilités de Saint-Flour XXXIII - 2003, Lecture Notes in Mathematics, 1896, Springer, 2007, pp. 147-181 | Zbl

[21] Philip E. Protter Stochastic differential equations, Stochastic integration and differential equations, Volume 21, Springer, 2005, pp. 243-355 | DOI | MR

[22] Patricia Reynaud-Bouret Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities, Probab. Theory Relat. Fields, Volume 126 (2003) no. 1, pp. 103-153 | DOI | MR | Zbl

[23] Patricia Reynaud-Bouret Compensator and exponential inequalities for some suprema of counting processes, Stat. Probab. Lett., Volume 76 (2006) no. 14, pp. 1514-1521 | DOI | MR | Zbl

[24] Alexander Sokol; Niels Hansen Exponential Martingales and Changes of Measure for Counting Processes, Stochastic Anal. Appl., Volume 33 (2012) no. 5, pp. 823-843 | DOI | MR | Zbl

[25] Michel Talagrand New concentration inequalities in product spaces, Invent. Math., Volume 126 (1996) no. 3, pp. 505-563 | DOI | MR | Zbl

[26] Sara Van De Geer Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes, Ann. Stat., Volume 23 (1995) no. 5, pp. 1779-1801 | MR | Zbl

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Closedness results for BMO semi-martingales and application to quadratic BSDEs

Pauline Barrieu; Nicolas Cazanave; Nicole El Karoui

C. R. Math (2008)

Reflected backward doubly stochastic differential equations driven by a Lévy process

Yong Ren

C. R. Math (2010)

A Bismut type formula for the Hessian of heat semigroups

Marc Arnaudon; Holger Plank; Anton Thalmaier

C. R. Math (2003)