Dans ce travail, nous considérons un processus de naissance et de mort de générateur et de probabilité invariante réversible . Étant données une fonction strictement croissante , et la norme lipschitzienne par rapport à , nous trouvons une représentation explicite de . En guise d’une application typique, nous retrouvons une formule variationnelle de M. F. Chen pour le trou spectral de dans . De plus, par la décomposition des martingales progressive-rétrogrades de Lyons-Zheng, nous obtenons des inégalités de concentration convexe pour des fonctionnelles additives de processus de naissance et de mort.
In this paper, we consider a birth-death process with generator and reversible invariant probability . Given an increasing function and the associated Lipschitz norm , we find an explicit formula for . As a typical application, with spectral theory, we revisit one variational formula of M. F. Chen for the spectral gap of in . Moreover, by Lyons-Zheng’s forward-backward martingale decomposition theorem, we get convex concentration inequalities for additive functionals of birth-death processes.
Mots clés : Birth-death process, spectral gap, Lipschitz function, Poisson equation, convex concentration inequality
@article{AIHPB_2009__45_1_58_0, author = {Liu, Wei and Ma, Yutao}, title = {Spectral gap and convex concentration inequalities for birth-death processes}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {58--69}, publisher = {Gauthier-Villars}, volume = {45}, number = {1}, year = {2009}, doi = {10.1214/07-AIHP149}, mrnumber = {2500228}, zbl = {1172.60023}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP149/} }
TY - JOUR AU - Liu, Wei AU - Ma, Yutao TI - Spectral gap and convex concentration inequalities for birth-death processes JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2009 SP - 58 EP - 69 VL - 45 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP149/ DO - 10.1214/07-AIHP149 LA - en ID - AIHPB_2009__45_1_58_0 ER -
%0 Journal Article %A Liu, Wei %A Ma, Yutao %T Spectral gap and convex concentration inequalities for birth-death processes %J Annales de l'I.H.P. Probabilités et statistiques %D 2009 %P 58-69 %V 45 %N 1 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP149/ %R 10.1214/07-AIHP149 %G en %F AIHPB_2009__45_1_58_0
Liu, Wei; Ma, Yutao. Spectral gap and convex concentration inequalities for birth-death processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 1, pp. 58-69. doi : 10.1214/07-AIHP149. http://www.numdam.org/articles/10.1214/07-AIHP149/
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