Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Intertwinings and Stein’s magic factors for birth–death processes

Bertrand Cloez and Claire Delplancke

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This article investigates second order intertwinings between semigroups of birth–death processes and discrete gradients on $\mathbb{N}$. It goes one step beyond a recent work of Chafaï and Joulin which establishes and applies to the analysis of birth–death semigroups a first order intertwining. Similarly to the first order relation, the second order intertwining involves birth–death and Feynman–Kac semigroups and weighted gradients on $\mathbb{N}$, and can be seen as a second derivative relation. As our main application, we provide new quantitative bounds on the Stein factors of discrete distributions. To illustrate the relevance of this approach, we also derive approximation results for the mixture of Poisson and geometric laws.


Cet article établit l’existence d’entrelacements au second ordre entre semi-groupes relatifs aux processus de naissance-mort et gradients discrets sur $\mathbb{N}$, allant ainsi un pas plus loin que les travaux récents de Chafaï et Joulin, qui concernent les entrelacements au premier ordre et leur application à l’analyse des semi-groupes de naissance-mort. Comme la relation du premier ordre, l’entrelacement de second ordre fait intervenir des semi-groupes de naissance-mort et de Feynman–Kac et des gradients à poids sur $\mathbb{N}$, et peut s’interpréter comme une relation de dérivation à l’ordre deux. Comme application principale, nous établissons des nouvelles bornes sur les facteurs de Stein relatifs aux distributions discrètes, et nous donnons également des résultats d’approximation pour le mélange de lois géométriques et le mélange de lois de Poisson, qui illustrent la pertinence de notre approche.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 341-377.

Received: 13 October 2016
Revised: 13 December 2017
Accepted: 15 January 2018
First available in Project Euclid: 18 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 47D08: Schrödinger and Feynman-Kac semigroups 60E05: Distributions: general theory 60F05: Central limit and other weak theorems

Birth–death processes Feynman–Kac semigroups Intertwinings Stein’s factors Stein’s method Distances between probability distributions


Cloez, Bertrand; Delplancke, Claire. Intertwinings and Stein’s magic factors for birth–death processes. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 341--377. doi:10.1214/18-AIHP884.

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