## Annals of Applied Probability

### Effective Berry–Esseen and concentration bounds for Markov chains with a spectral gap

Benoît Kloeckner

#### Abstract

Applying quantitative perturbation theory for linear operators, we prove nonasymptotic bounds for Markov chains whose transition kernel has a spectral gap in an arbitrary Banach algebra of functions $\mathscr{X}$. The main results are concentration inequalities and Berry–Esseen bounds, obtained assuming neither reversibility nor “warm start” hypothesis: the law of the first term of the chain can be arbitrary. The spectral gap hypothesis is basically a uniform $\mathscr{X}$-ergodicity hypothesis, and when $\mathscr{X}$ consist in regular functions this is weaker than uniform ergodicity. We show on a few examples how the flexibility in the choice of function space can be used. The constants are completely explicit and reasonable enough to make the results usable in practice, notably in MCMC methods.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1778-1807.

Dates
Revised: July 2018
First available in Project Euclid: 19 February 2019

https://projecteuclid.org/euclid.aoap/1550566842

Digital Object Identifier
doi:10.1214/18-AAP1438

Mathematical Reviews number (MathSciNet)
MR3914556

Zentralblatt MATH identifier
07057466

#### Citation

Kloeckner, Benoît. Effective Berry–Esseen and concentration bounds for Markov chains with a spectral gap. Ann. Appl. Probab. 29 (2019), no. 3, 1778--1807. doi:10.1214/18-AAP1438. https://projecteuclid.org/euclid.aoap/1550566842

#### References

• [1] Baladi, V. (2000). Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics 16. World Scientific, River Edge, NJ.
• [2] Bolthausen, E. (1982). The Berry–Esseén theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 283–289.
• [3] Bruin, H. and Todd, M. (2008). Equilibrium states for interval maps: Potentials with $\mathop{\mathrm{sup}}\phi-\inf\phi<h_{\mathrm{top}}(f)$. Comm. Math. Phys. 283 579–611.
• [4] Castro, A. and Varandas, P. (2013). Equilibrium states for non-uniformly expanding maps: Decay of correlations and strong stability. Ann. Inst. H. Poincaré Anal. Non Linéaire 30 225–249.
• [5] Chazottes, J.-R. and Gouëzel, S. (2012). Optimal concentration inequalities for dynamical systems. Comm. Math. Phys. 316 843–889.
• [6] Coelho, Z. and Parry, W. (1990). Central limit asymptotics for shifts of finite type. Israel J. Math. 69 235–249.
• [7] Cyr, V. and Sarig, O. (2009). Spectral gap and transience for Ruelle operators on countable Markov shifts. Comm. Math. Phys. 292 637–666.
• [8] Dedecker, J. and Fan, X. (2015). Deviation inequalities for separately Lipschitz functionals of iterated random functions. Stochastic Process. Appl. 125 60–90.
• [9] Dedecker, J. and Gouëzel, S. (2015). Subgaussian concentration inequalities for geometrically ergodic Markov chains. Electron. Commun. Probab. 20 no. 64, 12.
• [10] Dubois, L. (2011). An explicit Berry–Esséen bound for uniformly expanding maps on the interval. Israel J. Math. 186 221–250.
• [11] Erdős, P. (1939). On a family of symmetric Bernoulli convolutions. Amer. J. Math. 61 974–976.
• [12] Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Vol. II. Wiley, New York.
• [13] Giulietti, P., Kloeckner, B., Lopes, A. O. and Marcon, D. (2018). The calculus of thermodynamical formalism. J. Eur. Math. Soc. (JEMS) 20 2357–2412.
• [14] Glynn, P. W. and Ormoneit, D. (2002). Hoeffding’s inequality for uniformly ergodic Markov chains. Statist. Probab. Lett. 56 143–146.
• [15] Gómez, D. M. and Dartnell, P. (2012). Simple Monte Carlo integration with respect to Bernoulli convolutions. Appl. Math. 57 617–626.
• [16] Gouëzel, S. (2015). Limit theorems in dynamical systems using the spectral method. In Hyperbolic Dynamics, Fluctuations and Large Deviations. Proc. Sympos. Pure Math. 89 161–193. Amer. Math. Soc., Providence, RI.
• [17] Hennion, H. and Hervé, L. (2001). Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes in Math. 1766. Springer, Berlin.
• [18] Joulin, A. and Ollivier, Y. (2010). Curvature, concentration and error estimates for Markov chain Monte Carlo. Ann. Probab. 38 2418–2442.
• [19] Keller, G. and Liverani, C. (1999). Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 28 141–152.
• [20] Kloeckner, B. R. (2017). Effective perturbation theory for simple isolated eigenvalues of linear operators. J. Operator Theory To appear. Available at arXiv:1703.09425.
• [21] Kloeckner, B. R. (2017). Effective high-temperature estimates for intermittent maps. Ergodic Theory Dynam. Systems To appear. Available at arXiv:1704.00586.
• [22] Kloeckner, B. R. (2017). An optimal transportation approach to the decay of correlations for non-uniformly expanding maps. Ergodic Theory Dynam. Systems. To appear. Available at arXiv:1711.08052.
• [23] Kloeckner, B. R. (2018). Toy examples for effective concentration bounds.
• [24] Kontoyiannis, I., Lastras-Montano, L. A. and Meyn, S. P. (2005). Relative entropy and exponential deviation bounds for general Markov chains. In International Symposium on Information Theory, 2005 1563–1567. IEEE Press, New York.
• [25] Kontoyiannis, I. and Meyn, S. P. (2012). Geometric ergodicity and the spectral gap of non-reversible Markov chains. Probab. Theory Related Fields 154 327–339.
• [26] Lezaud, P. (1998). Chernoff-type bound for finite Markov chains. Ann. Appl. Probab. 8 849–867.
• [27] Lezaud, P. (2001). Chernoff and Berry–Esséen inequalities for Markov processes. ESAIM Probab. Stat. 5 183–201.
• [28] Liverani, C. (2001). Rigorous numerical investigation of the statistical properties of piecewise expanding maps. A feasibility study. Nonlinearity 14 463–490.
• [29] Nagaev, S. V. (1957). Some limit theorems for stationary Markov chains. Teor. Veroyatn. Primen. 2 389–416.
• [30] Nagaev, S. V. (1961). More exact limit theorems for homogeneous Markov chains. Teor. Veroyatn. Primen. 6 67–86.
• [31] Paulin, D. (2015). Concentration inequalities for Markov chains by Marton couplings and spectral methods. Electron. J. Probab. 20 no. 79, 32.
• [32] Paulin, D. (2016). Mixing and concentration by Ricci curvature. J. Funct. Anal. 270 1623–1662.
• [33] Peres, Y., Schlag, W. and Solomyak, B. (2000). Sixty years of Bernoulli convolutions. In Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998). Progress in Probability 46 39–65. Birkhäuser, Basel.
• [34] Roberts, G. O. and Rosenthal, J. S. (2004). General state space Markov chains and MCMC algorithms. Probab. Surv. 1 20–71.
• [35] Ruelle, D. (2004). Thermodynamic Formalism: The Mathematical Structures of Equilibrium Statistical Mechanics, 2nd ed. Cambridge Mathematical Library. Cambridge Univ. Press, Cambridge.
• [36] Solomyak, B. (1995). On the random series $\sum\pm\lambda^{n}$ (an Erdős problem). Ann. of Math. (2) 142 611–625.
• [37] Tyurin, I. S. (2011). Improvement of the remainder in the Lyapunov theorem. Teor. Veroyatn. Primen. 56 808–811.
• [38] Watanabe, S. and Hayashi, M. (2017). Finite-length analysis on tail probability for Markov chain and application to simple hypothesis testing. Ann. Appl. Probab. 27 811–845.