Skip to main content
SpringerLink
Log in

Local limit theorems for sums of independent random vectors

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

Abstract

Let pn(x) and qn(x) be the densities of the n-fold convolutions of the distributions F and G, respectively. One proves estimates for\(\mathop {\sup }\limits_x /\rho _n (x) - q_n (x)|\), expressed in terms of moment characteristics of F — G, under certain restrictions on the densities of F and G. Similar problems are solved for two lattice distributions and for a lattice and a continuous distribution.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Reading (1954).

    Google Scholar 

  2. V. M. Zolotarev, One-Dimensional Stable Distributions, Am. Math. Soc., Providence (1986).

    Google Scholar 

  3. Yu. V. Prokhorov and Yu. A. Rozanov, Probability Theory [in Russian], Nauka, Moscow (1973).

    Google Scholar 

  4. V. V. Petrov, Sums of Independent Random Variables, Springer, New York (1975).

    Google Scholar 

  5. R. N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions, Wiley, New York (1976).

    Google Scholar 

  6. G. M. Fikhtengol'ts, A Course of Differential and Integral Calculus, Vol. III [in Russian], Nauka, Moscow (1969).

    Google Scholar 

  7. A. Bikyalis, “Inequalities for multidimensional characteristic functions,” Litov. Mat. Sb. (Liet. Mat. Rinkinys),10, No. 4, 5–11 (1970).

    Google Scholar 

  8. L. Saulis, “The local limit theorem for the densities of distributions in Rk,” Litov. Mat. Sb. (Liet. Mat. Rinkinys),12, No. 4, 195–205 (1972).

    Google Scholar 

  9. L. V. Rozovskii, “On a multidimensional local limit theorem for the case of convergence to the normal law,” Litov. Mat. Sb. (Liet. Mat. Rinkinys),18, No. 1, 169–179 (1978).

    Google Scholar 

  10. I. Dubinskaite and A. Karoblis, “Local limit theorems for sums of identically distribuuted random vectors. III,” Litov. Mat. Sb. (Liet. Mat. Rinkinys),16, No. 4, 113–119 (1976).

    Google Scholar 

  11. A. Karoblis, “Local limit theorems for sums of identically distributed random vectors. I,” Litov. Mat. Sb. (Liet. Mat. Rinkinys),13, No. 3, 101–112 (1973).

    Google Scholar 

  12. I. I. Banis, “Estimation of the convergence rate in a local theorem in the multidimensional case,” Litov. Mat. Sb. (Liet. Mat. Rinkinys),19, No. 2, 13–21 (1979).

    Google Scholar 

  13. W. Hoeffding, “On sequences of sums of independent random vectors,” in: Proc. 4th Berkeley Symposium on Mathematical Statistics and Probability, 1960, Vol. II, Univ. of California Press, Berkeley (1961), pp. 213–226.

    Google Scholar 

Download references

Authors
  1. A. I. Karoblis
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 193–212, 1986.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Karoblis, A.I. Local limit theorems for sums of independent random vectors. J Math Sci 38, 2279–2287 (1987). https://doi.org/10.1007/BF01093829

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01093829

Keywords

Search

Navigation