zbMATH — the first resource for mathematics

Asymptotic development for the CLT in total variation distance. (English) Zbl 1346.60016
Authors’ abstract: The aim of this paper is to study the asymptotic expansion in the total variation distance in the central limit theorem when the law of the basic random variable is locally lower-bounded by the Lebesgue measure (or equivalently, has an absolutely continuous component): we develop the error in powers of \(n^{-1/2}\) and give an explicit formula for the approximating measure.

60F05 Central limit and other weak theorems
60H07 Stochastic calculus of variations and the Malliavin calculus
60H05 Stochastic integrals
60B10 Convergence of probability measures
60G50 Sums of independent random variables; random walks
PDF BibTeX Cite
Full Text: DOI Euclid
[1] Abramowitz, M. and Stegun, C.A. (1972). Bernoulli and Euler polynomials and the Euler-Maclaurin formula. In Handbook of Mathematical Functions with Formulas , Graphs , and Mathematical Tables 9th ed. New York: Dover.
[2] Bakry, D., Gentil, I. and Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 348 . Cham: Springer. · Zbl 1376.60002
[3] Bally, V. and Caramellino, L. (2014). On the distances between probability density functions. Electron. J. Probab. 19 1-33. · Zbl 1307.60072
[4] Bally, V. and Clément, E. (2011). Integration by parts formula and applications to equations with jumps. Probab. Theory Related Fields 151 613-657. · Zbl 1243.60045
[5] Bhattacharaya, R.N. and Ranga Rao, R. (2010). Normal Approximation and Asymptotic Expansions. SIAM Classics in Applied Mathematics 64 . Philadelphia: SIAM.
[6] Bhattacharya, R.N. (1968). Berry-Esseen bounds for the multi-dimensional central limit theorem. Bull. Amer. Math. Soc. 74 285-287. · Zbl 0179.48102
[7] Bobkov, S.G., Chistyakov, G.P. and Götze, F. (2014). Berry-Esseen bounds in the entropic central limit theorem. Probab. Theory Related Fields 159 435-478. · Zbl 1307.60011
[8] Dudley, R.M. (2002). Real Analysis and Probability. Cambridge Studies in Advanced Mathematics 74 . Cambridge: Cambridge Univ. Press. · Zbl 1023.60001
[9] Nourdin, I., Nualart, D. and Poly, G. (2013). Absolute continuity and convergence of densities for random vectors on Wiener chaos. Electron. J. Probab. 18 19 pp. · Zbl 1285.60053
[10] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus : From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192 . Cambridge: Cambridge Univ. Press. · Zbl 1266.60001
[11] Nourdin, I. and Poly, G. (2013). Convergence in total variation on Wiener chaos. Stochastic Process. Appl. 123 651-674. · Zbl 1259.60029
[12] Nourdin, I. and Poly, G. (2015). An invariance principle under the total variation distance. Stochastic Process. Appl. 125 2190-2205. · Zbl 1321.60065
[13] Nualart, D. (2006). The Malliavin Calculus and Related Topics , 2nd ed. Probability and Its Applications ( New York ). Berlin: Springer. · Zbl 1099.60003
[14] Prohorov, Yu.V. (1952). A local theorem for densities. Doklady Akad. Nauk SSSR ( N.S. ) 83 797-800.
[15] Ranga Rao, R. (1961). On the central limit theorem in \(R_{k}\). Bull. Amer. Math. Soc. 67 359-361. · Zbl 0099.13003
[16] Sirazhdinov, S.Kh. and Mamatov, M. (1962). On convergence in the mean for densities. Theory Probab. Appl. 7 424-428. · Zbl 0302.60015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.