# Article

Full entry | PDF   (1.2 MB)
Keywords:
sum of random vectors; the total variation distance; bound of closeness; Zolotarev’s metric; characteristic function
Summary:
Let $(X_n, n\ge 1), (\tilde{X}_n, n\ge 1)$ be two sequences of i.i.d. random vectors with values in ${\mathbb{R}}^k$ and $S_n=X_1+\cdots +X_n$, $\tilde{S}_n=\tilde{X}_1+\cdots +\tilde{X}_n$, $n\ge 1$. Assuming that $EX_1=E\tilde{X}_1$, $E|X_1|^2<\infty$, $E|\tilde{X}_1|^{k+2}<\infty$ and the existence of a density of $\tilde{X}_1$ satisfying the certain conditions we prove the following inequalities: $v(S_n,\tilde{S}_n)\le c\;\max \big \lbrace v(X_1,\tilde{X}_1), \zeta _2(X_1,\tilde{X}_1)\big \rbrace , \quad n=1,2,\dots ,$ where $v$ and $\zeta _2$ are the total variation and Zolotarev’s metrics, respectively.
References:
[1] Araujo A., Giné E.: The Central Limit Theorem for Real and Banach Valued Random Variables. Wiley, New York 1980 MR 0576407 | Zbl 0457.60001
[2] Asmussen S.: Applied Probability and Queues. Wiley, Chichester 1987 MR 0889893 | Zbl 1029.60001
[3] Bhattacharya R. N., Rao R. Ranga: Normal Approximation and Asymptotic Expansions. Wiley, New York 1976 MR 0436272
[4] Dudley R. M.: Uniform Central Limit Theorems. Cambridge University Press, Cambridge 1999 MR 1720712 | Zbl 1139.60016
[5] Gordienko E. I.: Estimates of stability of geometric convolutions. Appl. Math. Lett. 12 (1999), 103–106 DOI 10.1016/S0893-9659(99)00064-6 | MR 1750146 | Zbl 0944.60035
[6] Gordienko E. I., Chávez J. Ruiz de: New estimates of continuity in $M|GI|1|\infty$ queues. Queueing Systems Theory Appl. 29 (1998), 175–188 MR 1654484
[7] Grandell J.: Aspects of Risk Theory. Springer–Verlag, Heidelberg 1991 MR 1084370 | Zbl 0717.62100
[8] Kalashnikov V.: Geometric Sums: Bounds for Rare Events with Applications. Kluwer Academic Publishers, Dordrecht 1997 MR 1471479 | Zbl 0881.60043
[9] Kalashnikov V., Konstantinidis D.: The ruin probability. Fund. Appl. Math. 2 (1996), 1055–1100 (in Russian) MR 1785772
[10] Prokhorov A. V., Ushakov N. G.: On the problem of reconstructing a summands distribution by the distribution of their sum. Theory Probab. Appl. 46 (2002), 420–430 DOI 10.1137/S0040585X97979202 | MR 1978662 | Zbl 1032.60010
[11] Senatov V. V.: Uniform estimates of the rate of convergence in the multi-dimensional central limit theorem. Theory Probab. Appl. 25 (1980), 745–759
[12] Senatov V. V.: Qualitative effects in estimates for the rate of convergence in the central limit theorem in multidimensional spaces. Proc. Steklov Inst. Math. 215 (1996), 4, 1–237 MR 1632100
[13] Zhukov, Yu. V.: On the accuracy of normal approximation for the densities of sums of independent identically distributed random variables. Theory Probab. Appl. 44 (2000), 785–793 MR 1811136 | Zbl 0967.60022
[14] Zolotarev V.: Ideal metrics in the problems of probability theory. Austral. J. Statist. 21 (1979), 193–208 DOI 10.1111/j.1467-842X.1979.tb01139.x | MR 0561947 | Zbl 0428.62012

Partner of