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Teor. Veroyatnost. i Primenen., 1965, Volume 10, Issue 3, Pages 519–526 (Mi tvp547)  

This article is cited in 9 scientific papers (total in 9 papers)

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On the closeness of the distributions of the two sums of independent random variables

V. M. Zolotarev


Abstract: Let $\{\xi_j\}$, $j=1,2,…,n$ (resp. $\{\eta_j\}$, $j=1,2,…,n$) be independent random variables with distribution functions $\{F_j\}$, $j=1,2,…,n$ (resp. $\{G_j\}$, $j=1,2,…,n$) and let $F$ (resp. $G$) be the distribution function of the sum $\xi=\xi_1+…+\xi_n$ (resp. $\eta=\eta_1+…+\eta_n$).
Let us denote
$$ \mu(k)=\sum_{j=1}^n|\int x^kd(F_j-G_j)|,\quad \nu(r)=\sum_{j=1}^n\int|x|^r|d(F_j-G_j)|. $$
We suppose that $\mu(0)=\mu(1)=…=\mu(m)=0$ and $\nu(r)$ exist for some $r$, $m\le r\le m+1$. In this case
a) if the distribution of $\eta$ has a density bounded by a constant $q$, then
$$ |F(x)-G(x)|<C[\nu(r)q^r]^\frac1{1+r},\eqno{(\text*)} $$

b) if $F$ and $G$ are lattice distributions with the same points of discontinuity and the same largest common factor of the length of the intervals between jumps $h$, then
$$ |F(x)-G(x)|<C_1[\nu(r)h^{-r}]\eqno{(**)} $$
where $C$ and $C_1$ are constants depending only on $m$ and $r$.
In the case a) an estimation of the type (**), which is better then one of the type (*) can be achieved only when some additional requirements on $\xi_j$ are satisfied. The estimations (*) and (**) make it possible to formulate some sufficient conditions for $F$ to converge to infinitely divisible distribution $G$ when the summands $\xi_j$ are not necessarily uniformly infinitesimal.

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English version:
Theory of Probability and its Applications, 1965, 10:3, 472–479

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Received: 10.05.1965

Citation: V. M. Zolotarev, “On the closeness of the distributions of the two sums of independent random variables”, Teor. Veroyatnost. i Primenen., 10:3 (1965), 519–526; Theory Probab. Appl., 10:3 (1965), 472–479

Citation in format AMSBIB
\by V.~M.~Zolotarev
\paper On the closeness of the distributions of the two sums of independent random variables
\jour Teor. Veroyatnost. i Primenen.
\yr 1965
\vol 10
\issue 3
\pages 519--526
\jour Theory Probab. Appl.
\yr 1965
\vol 10
\issue 3
\pages 472--479

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    This publication is cited in the following articles:
    1. V. I. Rotar', “On summation of independent variables in a non-classical situation”, Russian Math. Surveys, 37:6 (1982), 151–175  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. V. Yu. Korolev, I. G. Shevtsova, “An upper estimate for the absolute constant in the Berry–Esseen inequality”, Theory Probab. Appl., 54:4 (2010), 638–658  mathnet  crossref  crossref  mathscinet  isi
    3. Tyurin I.S., “On the accuracy of the Gaussian approximation”, Doklady Mathematics, 80:3 (2009), 840–843  mathnet  mathnet  crossref  mathscinet  zmath  isi
    4. Paulauskas V., “On the rate of convergence to bivariate stable laws”, Lithuanian Mathematical Journal, 49:4 (2009), 426–445  crossref  mathscinet  zmath  isi
    5. I. G. Shevtsova, “Nekotorye otsenki dlya kharakteristicheskikh funktsii s primeneniem k utochneniyu neravenstva Mizesa”, Inform. i ee primen., 3:3 (2009), 69–78  mathnet
    6. I. S. Tyurin, “On the convergence rate in Lyapunov's theorem”, Theory Probab. Appl., 55:2 (2011), 253–270  mathnet  crossref  crossref  mathscinet  isi
    7. Korolev V. Shevtsova I., “An Improvement of the Berry-Esseen Inequality with Applications to Poisson and Mixed Poisson Random Sums”, Scand. Actuar. J., 2012, no. 2, 81–105  crossref  isi
    8. Shevtsova I., “On the Accuracy of the Approximation of the Complex Exponent by the First Terms of its Taylor Expansion with Applications”, J. Math. Anal. Appl., 418:1 (2014), 185–210  crossref  mathscinet  isi  elib
    9. Bobkov S.G., “Asymptotic Expansions For Products of Characteristic Functions Under Moment Assumptions of Non-Integer Orders”, Convexity and Concentration, IMA Volumes in Mathematics and Its Applications, 161, ed. Carlen E. Madiman M. Werner E., Springer, 2017, 297–357  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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