logo CRAS
Comptes Rendus. Mathématique
Probability theory
Exponential inequalities for the supremum of some counting processes and their square martingales
Comptes Rendus. Mathématique, Volume 359 (2021) no. 8, pp. 969-982.

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.

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.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/crmath.206
Classification: 60G55
Ronan Le Guével 1

1. Université Rennes 2, France.
@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
DA  - 2021///
SP  - 969
EP  - 982
VL  - 359
IS  - 8
PB  - Académie des sciences, Paris
UR  - https://doi.org/10.5802/crmath.206
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
%U https://doi.org/10.5802/crmath.206
%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 1235426 | Zbl 0789.60031

[2] Richard F. Bass The measurability of hitting times, Electron. Commun. Probab., Volume 15 (2010), pp. 99-105 | Article | MR 2606507 | Zbl 1202.60054

[3] Richard F. Bass Stochastic processes, Cambridge Series in Statistical and Probabilistic Mathematics, 33, Cambridge University Press, 2011 | Zbl 1247.60001

[4] Bernard Bercu; Bernard Delyon; Emmanuel Rio Concentration Inequalities for Sums and Martingales, SpringerBriefs in Mathematics, Springer, 2015 | Zbl 1337.60002

[5] Pierre Brémaud Point processes and queues: martingale dynamics, Springer Series in Statistics, Springer, 1981 | Zbl 0478.60004

[6] Pierre Brémaud; Laurent Massoulié Stability of nonlinear Hawkes processes, Ann. Probab., Volume 24 (1996) no. 3, pp. 1563-1588 | MR 1411506 | Zbl 0870.60043

[7] Jean Bretagnolle A new large deviation inequality for U-statistics of order 2, ESAIM, Probab. Stat., Volume 3 (1999), pp. 151-162 | Article | Numdam | MR 1742613 | Zbl 0957.60031

[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 2281881 | Zbl 1096.62040

[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 2779402 | Zbl 1207.62161

[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 1857312 | Zbl 0969.60024

[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 0787.60020

[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 | Article | MR 279864 | Zbl 0216.22203

[13] Wassily Hoeffding Probability Inequalities for Sums of Bounded Random Variables, J. Am. Stat. Assoc., Volume 58 (1963) no. 301, pp. 13-30 | Article | MR 144363 | Zbl 0127.10602

[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 1036.60015

[15] Olav Kallenberg Foundations of Modern Probability, Probability and Its Applications, Springer, 1997 | Zbl 0892.60001

[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 1457627 | Zbl 0895.60018

[17] Béatrice Laurent Adaptive estimation of a quadratic functional of a density by model selection, ESAIM, Probab. Stat., Volume 9 (2005), pp. 1-18 | Article | Numdam | MR 2148958 | Zbl 1136.62333

[18] Michel Ledoux On Talagrand’s deviation inequalities for product measures, ESAIM, Probab. Stat., Volume 1 (1997), pp. 63-87 | Article | Numdam | MR 1399224 | Zbl 0869.60013

[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 1782276 | Zbl 1140.60310

[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 1170.60006

[21] Philip E. Protter Stochastic differential equations, Stochastic integration and differential equations, Volume 21, Springer, 2005, pp. 243-355 | Article | MR 2273672

[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 | Article | MR 1981635 | Zbl 1019.62079

[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 | Article | MR 2245573 | Zbl 1101.60033

[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 | Article | MR 3378040 | Zbl 1325.60063

[25] Michel Talagrand New concentration inequalities in product spaces, Invent. Math., Volume 126 (1996) no. 3, pp. 505-563 | Article | MR 1419006 | Zbl 0893.60001

[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 1370307 | Zbl 0852.60019

Cited by Sources: