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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

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

Short Communications

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

V. M. Zolotarev

Moscow

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

Bibliographic databases:

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
\Bibitem{Zol65} \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 \mathnet{http://mi.mathnet.ru/tvp547} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=189109} \zmath{https://zbmath.org/?q=an:0214.17402} \transl \jour Theory Probab. Appl. \yr 1965 \vol 10 \issue 3 \pages 472--479 \crossref{https://doi.org/10.1137/1110055} 

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1. V. I. Rotar', “On summation of independent variables in a non-classical situation”, Russian Math. Surveys, 37:6 (1982), 151–175
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3. Tyurin I.S., “On the accuracy of the Gaussian approximation”, Doklady Mathematics, 80:3 (2009), 840–843
4. Paulauskas V., “On the rate of convergence to bivariate stable laws”, Lithuanian Mathematical Journal, 49:4 (2009), 426–445
5. I. G. Shevtsova, “Nekotorye otsenki dlya kharakteristicheskikh funktsii s primeneniem k utochneniyu neravenstva Mizesa”, Inform. i ee primen., 3:3 (2009), 69–78
6. I. S. Tyurin, “On the convergence rate in Lyapunov's theorem”, Theory Probab. Appl., 55:2 (2011), 253–270
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
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
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