Electronic Journal of Probability

A Local Limit Theorem for sums of independent random vectors

Dmitry Dolgopyat

Full-text: Open access


We prove a local limit theorem for sums of independent random vectors satisfying appropriate tightness assumptions. In particular, the local limit theorem holds in dimension 1 if the summands are uniformly bounded.

Article information

Electron. J. Probab., Volume 21 (2016), paper no. 39, 15 pp.

Received: 11 April 2015
Accepted: 17 May 2016
First available in Project Euclid: 15 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization

local limit theorem characteristic function lattice distribution concentration inequality

Creative Commons Attribution 4.0 International License.


Dolgopyat, Dmitry. A Local Limit Theorem for sums of independent random vectors. Electron. J. Probab. 21 (2016), paper no. 39, 15 pp. doi:10.1214/16-EJP4232. https://projecteuclid.org/euclid.ejp/1465991837

Export citation


  • [1] Davis B., McDonald D. An elementary proof of the local central limit theorem, J. Theoret. Probab. 8 (1995) 693–701.
  • [2] Doney R. A. A bivariate local limit theorem, J. Multivariate Anal. 36 (1991) 95–102.
  • [3] Feller W. An introduction to probability theory and its applications, Vol. II. 2d ed. John Wiley & Sons, New York-London-Sydney (1971) xxiv+669 pp.
  • [4] Gamkrelidze N. G. On a local limit theorem for integer random vectors, Theory Probab. Appl. 59 (2015) 494–499.
  • [5] Giuliano R., Weber M. Local limit theorems in some random models from number theory, preprint arXiv:1502.05939.
  • [6] Ibragimov I. A., Linnik Yu. V. Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen (1971) 443 pp.
  • [7] Lebowitz J. L., Pittel B., Ruelle D., Speer E. Central limit theorems, Lee-Yang zeros, and graph-counting polynomials, J. Combin. Th. 141 (2016) 147–183.
  • [8] Maller R. A. A local limit theorem for independent random variables, Stoch. Process. Appl. 7 (1978) 101–111.
  • [9] Petrov V. V. Limit theorems of probability theory. Sequences of independent random variables, Oxford Studies in Probability 4 Oxford University Press, New York, 1995. xii+292 pp.
  • [10] Prokhorov Yu. V. On the local limit theorem for lattice distributions, Dokl. Akad. Nauk SSSR 98 (1954) 535–538.
  • [11] Rollin A., Ross N. Local limit theorems via Landau-Kolmogorov inequalities, Bernoulli 21 (2015) 851–880.
  • [12] Rozanov Yu. A. On a local limit theorem for lattice distributions, Theory Probab. Appl., 2 (1957) 260–265.