Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

Bertrand Cloez and Claire Delplancke

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

Permanent link to this document

Digital Object Identifier

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.

Export citation


  • [1] J. Abate, M. Kijima and W. Whitt. Decompositions of the M/M/$1$ transition function. Queueing Syst. 9 (3) (1991) 323–336.
  • [2] F. Baccelli and W. A. Massey. A sample path analysis of the M/M/$1$ queue. J. Appl. Probab. 26 (2) (1989) 418–422.
  • [3] D. Bakry and M. Émery. Diffusions hypercontractives. In Séminaire de probabilités, XIX, 1983/84 177–206. Lecture Notes in Math. 1123. Springer, Berlin, 1985.
  • [4] A. D. Barbour and T. C. Brown. Stein’s method and point process approximation. Stochastic Process. Appl. 43 (1) (1992) 9–31.
  • [5] A. D. Barbour, H. L. Gan and A. Xia. Stein factors for negative binomial approximation in Wasserstein distance. Bernoulli 21 (2) (2015) 1002–1013.
  • [6] A. D. Barbour, L. Holst and S. Janson. Poisson Approximation. Oxford Studies in Probability 2. The Clarendon Press, Oxford University Press, New York, 1992.
  • [7] A. D. Barbour and A. Xia. On Stein’s factors for Poisson approximation in Wasserstein distance. Bernoulli 12 (6) (2006) 943–954.
  • [8] M. Bonnefont and A. Joulin. Intertwining relations for one-dimensional diffusions and application to functional inequalities. Potential Anal. 41 (4) (2014) 1005–1031.
  • [9] T. C. Brown and M. J. Phillips. Negative binomial approximation with Stein’s method. Methodol. Comput. Appl. Probab. 1 (4) (1999) 407–421.
  • [10] T. C. Brown and A. Xia. Stein’s method and birth–death processes. Ann. Probab. 29 (3) (2001) 1373–1403.
  • [11] D. Chafaï and A. Joulin. Intertwining and commutation relations for birth–death processes. Bernoulli 19 (5A) (2013) 1855–1879.
  • [12] L. H. Y. Chen. Poisson approximation for dependent trials. Ann. Probab. 3 (3) (1975) 534–545.
  • [13] M.-F. Chen. Estimation of spectral gap for Markov chains. Acta Math. Sinica (N.S.) 12 (4) (1996) 337–360.
  • [14] M.-F. Chen. From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition. World Scientific Publishing Co., Inc., River Edge, NJ, 2004.
  • [15] B. Cloez. Wasserstein decay of one dimensional jump-diffusions, 2012. Available at arXiv:1202.1259.
  • [16] R. L. Dobrušin. On conditions of regularity of stationary Markov processes with a denumerable number of possible states. Uspehi Mat. Nauk (N.S.) 7 (6(52)) (1952) 185–191. In Russian.
  • [17] P. Eichelsbacher and G. Reinert. Stein’s method for discrete Gibbs measures. Ann. Appl. Probab. 18 (4) (2008) 1588–1618.
  • [18] D. G. Kendall. On some modes of population growth leading to R. A. Fisher’s logarithmic series distribution. Biometrika 35 (1948) 6–15.
  • [19] C. Ley, G. Reinert and Y. Swan. Stein’s method for comparison of univariate distributions. Probab. Surv. 14 (2017) 1–52.
  • [20] T. Lindvall. Lectures on the Coupling Method. Dover Publications, Inc., Mineola, NY, 2002. Corrected reprint of the 1992 original.
  • [21] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer, London, 1993.
  • [22] E. A. Peköz. Stein’s method for geometric approximation. J. Appl. Probab. 33 (3) (1996) 707–713.
  • [23] E. A. Peköz, A. Röllin and N. Ross. Total variation error bounds for geometric approximation. Bernoulli 19 (2) (2013) 610–632.
  • [24] M. J. Phillips. Stochastic process approximation and network applications. Ph.D. thesis, Univ. Melbourne, 1996.
  • [25] E. Rio. Distances minimales et distances idéales. C. R. Acad. Sci. Paris Sér. I Math. 326 (9) (1998) 1127–1130.
  • [26] H. Robbins. A remark on Stirling’s formula. Amer. Math. Monthly 62 (1955) 26–29.
  • [27] C. Stein. A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. II: Probability Theory 583–602. Univ. California Press, Berkeley, CA, 1972.
  • [28] A. Szulga. On minimal metrics in the space of random variables. Teor. Veroyatnost. i Primenen. 27 (2) (1982) 401–405.
  • [29] V. M. Zolotarev. Metric distances in spaces of random variables and their distributions. Mat. Sb. (N.S.) 101(143) (3) (1976) 416–454, 456.