Breuillard, E. Diophantine approximations and local limit theorem in \(\mathbb R^d\). (Distributions diophantiennes et théorème limite local sur \(\mathbb R^d\).) (French) Zbl 1079.60050 Probab. Theory Relat. Fields 132, No. 1, 39-73 (2005). Author’s abstract: We study the speed of convergence of \(n^{d/2} \int f \,d\mu^{*n}\) in the local limit theorem on \(\mathbb{R}^d\) under very general conditions upon the function \(f\) and the distribution \(\mu\). We show that this speed is at least of order \(1/n\) and we give a simple characterization (in Diophantine terms) of those measures for which this speed (and the full local Edgeworth expansion) holds for smooth enough \(f\). We then derive a uniform local limit theorem for moderate deviations under a mild moment assumption. This in turn yields other limit theorems when \(f\) is no longer assumed integrable but only bounded and Lipschitz or Hölder. We finally give an application to equidistribution of random walks. Reviewer: Bero Roos (Hamburg) Cited in 19 Documents MSC: 60G50 Sums of independent random variables; random walks 60F05 Central limit and other weak theorems Keywords:Edgeworth expansion; Diophantine conditions PDFBibTeX XMLCite \textit{E. Breuillard}, Probab. Theory Relat. Fields 132, No. 1, 39--73 (2005; Zbl 1079.60050) Full Text: DOI References: [1] Amosova, Lit. Mat. Sb., 14, 401 (3) · Zbl 0348.60029 [2] Breiman, L.: Probability. Addison-Wesley, (1968) · Zbl 0174.48801 [3] Carlsson, Compositio Math., 46, 227 (2) · Zbl 0492.60083 [4] Carlsson, Ann. Probab., 11, 143 (1) · Zbl 0507.60081 [5] Cramér, H.: Les sommes et les fonctions de variables aléatoires. Paris Hermann, (1938) · Zbl 0022.24104 [6] Cramér, H.: Elements of probability theory and some of its applications. Cambridge, (1962) [7] Feller, W.: An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, (1971) · Zbl 0219.60003 [8] Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Addison-Wesley, (1954) · Zbl 0056.36001 [9] Hinderer, Math. Nachr., 130, 225 (1987) · Zbl 0623.60105 [10] Höglund, T., A multi-dimensional renewal theorem, Bull. Sc. Math, 2ème série, 112, 111-138 (1988) · Zbl 0646.60090 [11] Keener, Stochastic Process. Appl., 34, 137 (1) · Zbl 0695.60085 [12] Michel, R.: Results on probabilities of moderate deviations. Ann. Prob. (2), 349-353 (1974) · Zbl 0282.60015 [13] Nagaev, Veroyatnost. i Primenen., 24, 565 (3) · Zbl 0408.60088 [14] Nagaev, Prob. Th. Appl., 14, 51 (1969) · Zbl 0196.21002 [15] Nagaev, Ann. Prob., 7, 745 (5) · Zbl 0418.60033 [16] et, Sankhya Ser. A, 27, 325 (1965) · Zbl 0178.53802 [17] Rudin, W.: Real and Complex Analysis, Addison-Wesley · Zbl 0142.01701 [18] Slastnikov, Probab. Theo. and Appl., 23, 325 (2) [19] Stam, Compositio Math., 21, 383 (1969) · Zbl 0192.54601 [20] Stone, Ann. Math. Stat, 36, 546 (1965) · Zbl 0135.19204 [21] Stone, Ch.: Ratio local and ratio limit theorems. 5th Berkeley Symp. on Math. Stat. and Prob. Berkeley and Los Angeles, UCP (1966), Vol II, 2, pp. 217-224 · Zbl 0236.60021 [22] Stone, Ch.: Application of unsmoothing and Fourier Analysis to random walks. Markov Processes and Potential Theory, Madison Wis, 165-192 (1967) · Zbl 0178.20101 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.