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Integral estimation based on Markovian design

  • Romain Azaïs (a1), Bernard Delyon (a2) and François Portier (a3)


Suppose that a mobile sensor describes a Markovian trajectory in the ambient space and at each time the sensor measures an attribute of interest, e.g. the temperature. Using only the location history of the sensor and the associated measurements, we estimate the average value of the attribute over the space. In contrast to classical probabilistic integration methods, e.g. Monte Carlo, the proposed approach does not require any knowledge of the distribution of the sensor trajectory. We establish probabilistic bounds on the convergence rates of the estimator. These rates are better than the traditional `root n'-rate, where n is the sample size, attached to other probabilistic integration methods. For finite sample sizes, we demonstrate the favorable behavior of the procedure through simulations and consider an application to the evaluation of the average temperature of oceans.


Corresponding author

* Postal address: Team BIGS, Inria Nancy ‒ Grand-Est Research Centre, 615 rue du Jardin Botanique, 54600 Villers-lès-Nancy, France.
** Postal address: CNRS, IRMAR - UMR 6625, University of Rennes 1, F-35000 Rennes, France. Email address:
*** Postal address: LTCI, Télécom ParisTech, 46 rue Barrault, 75634 Paris, Cedex 13, France.


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[1]Athreya, K. B. and Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245, 493501.
[2]Azaïs, R., Delyon, B. and Portier, F. (2018). Integral estimation based on Markovian design. Supplementary material. Available at
[3]Bertail, P. and Clémençon, S. (2011). A renewal approach to Markovian U-statistics. Math. Meth. Statist. 20, 79105.
[4]Chacón, J. E. and Duong, T. (2010). Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices. TEST 19, 375398.
[5]Delyon, B. and Portier, F. (2016). Integral approximation by kernel smoothing. Bernoulli 22, 21772208.
[6]Duong, T. (2007). Ks: kernel density estimation and kernel discriminant analysis for multivariate data in R. J. Statist. Software 21, 16pp.
[7]Einmahl, U. and Mason, D. M. (2005). Uniform in bandwidth consistency of kernel-type function estimators. Ann. Statist. 33, 13801403.
[8]Evans, M. and Swartz, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods. Oxford University Press.
[9]Gasser, T., Muller, H.-G. and Mammitzsch, V. (1985). Kernels for nonparametric curve estimation. J. R. Statist. Soc. B 47, 238252.
[10]Hansen, B. E. (2008). Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24, 726748.
[11]Härdle, W. and Stoker, T. M. (1989). Investigating smooth multiple regression by the method of average derivatives. J. Amer. Statist. Assoc. 84, 986995.
[12]Jarner, S. F. and Roberts, G. O. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Prob. 12, 224247.
[13]Kosaka, Y. and Xie, S.-P. (2013). Recent global-warming hiatus tied to equatorial Pacific surface cooling. Nature 501, 403407.
[14]Li, Q. and Racine, J. S. (2007). Nonparametric Econometrics: Theory and Practice. Princeton University Press.
[15]Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd edn. Cambridge University Press.
[16]Nolan, D. and Pollard, D. (1987). U-processes: rates of convergence. Ann. Statist. 15, 780799.
[17]Novak, E. (2016). Some results on the complexity of numerical integration. In Monte Carlo and Quasi-Monte Carlo Methods, Springer, Cham, pp. 161183.
[18]Nummelin, E. (1978). A splitting technique for Harris recurrent Markov chains. Z. Wahrscheinlichkeitsth. 43, 309318.
[19]Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge University Press.
[20]R Development Core Team (2007). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna. Available at
[21]Rahmstorf, S. et al. (2015). Exceptional twentieth-century slowdown in Atlantic Ocean overturning circulation. Nature Climate Change 5, 475480.
[22]Robert, C. P. and Casella, G. (2004). Monte Carlo Statistical Methods, 2nd edn. Springer, New York.
[23]Roget-Vial, C. (2003). Deux contributions à l'étude semi-paramétrique d'un modèle de régression. Doctoral thesis. University of Rennes 1.
[24]Roussas, G. G. (1969). Nonparametric estimation of the transition distribution function of a Markov process. Ann. Math. Statist. 40, 13861400.
[25]Roxy, M. K., Ritika, K., Terray, P. and Masson, S. (2014). The curious case of Indian Ocean warming. J. Climate 27, 85018509.
[26]Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis. Chapman & Hall, London.
[27]Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer, New York.
[28]Van der Vaart, A. W. (1998). Asymptotic Statistics. Cambridge University Press.
[29]Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.


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