Geodesic
Estimates of the concentration functions of weighted sums of independent random variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 57 (2012) no. 4, pp. 768-777 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $X_1,\dots,X_n$ be independent and identially distributed random variables. The paper deals with obtaining upper bounds on the concentration function of the weighted sums $\sum_{k=1}^na_kX_k$ based on the coefficients $a_k$, $1\leqslant k\leqslant n$. Results obtained in this paper improve over the recent works in [O. Friedland and S. Sodin, C. R., Math., Acad. Sci. Paris 345, No. 9, 513–518 (2007; Zbl 1138.60023)] and [M. Rudelson and R. Vershynin, Adv. Math. 218, No. 2, 600–633 (2008; Zbl 1139.15015), Commun. Pure Appl. Math. 62, No. 12, 1707–1739 (2009; Zbl 1183.15031)].
Keywords: concentration functions; inequalities; sums of independent random variables; Littlewood–Offord problem. 
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     title = {Estimates of the concentration functions of weighted sums of independent random variables},
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Yu. S. Eliseeva; A. Yu. Zaitsev. Estimates of the concentration functions of weighted sums of independent random variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 57 (2012) no. 4, pp. 768-777. http://geodesic.mathdoc.fr/item/TVP_2012_57_4_a8/

[1] Arak T. V., “O skorosti skhodimosti v ravnomernoi predelnoi teoreme Kolmogorova, I”, Teoriya veroyatn. i ee primen., 26:2 (1981), 225–245 | MR | Zbl

[2] Arak T. V., Zaitsev A. Yu., “Ravnomernye predelnye teoremy dlya summ nezavisimykh sluchainykh velichin”, Tr. MIAN, 174, 1986, 214 pp. | MR | Zbl

[3] Bretagnolle J., “Sur l'inégalité de concentration de Doeblin–Lévy, Rogozin–Kesten”, Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life, Birkhäuser, Boston, 2004, 533–551 | DOI | MR

[4] Esséen C.-G., “On the Kolmogorov–Rogozin inequality for the concentration function”, Z. Wahrscheinlichkeitstheor. verw. Geb., 5 (1966), 210–216 | DOI | MR | Zbl

[5] Esséen C.-G., “On the concentration function of a sum of independent random variables”, Z. Wahrscheinlichkeitstheor. verw. Geb., 9 (1968), 290–308 | DOI | MR | Zbl

[6] Friedland O., Sodin S., “Bounds on the concentration function in terms of Diophantine approximation”, C. R. Math. Acad. Sci. Paris, 345:9 (2007), 513–518 | DOI | MR | Zbl

[7] Khengartner V., Teodoresku R., Funktsii kontsentratsii, Nauka, M., 1980, 176 pp. | MR

[8] Kesten H., “A sharper form of the Doeblin–Lévy–Kolmogorov–Rogozin inequality for concentration functions”, Math. Scand., 25 (1969), 133–144 | MR | Zbl

[9] Miroshnikov A. L., Rogozin B. A., “Neravenstva dlya funktsii kontsentratsii”, Teoriya veroyatn. i ee primen., 25:1 (1980), 178–183 | MR | Zbl

[10] Nagaev S. V., Khodzhabagyan S. S., “Ob otsenke funktsii kontsentratsii summ nezavisimykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 41:3 (1996), 655–665 | MR | Zbl

[11] Nguyen Hoi, Vu Van, “Optimal inverse Littlewood–Offord theorems”, Adv. Math., 226:6 (2011), 5298–5319 | DOI | MR | Zbl

[12] Petrov V. V., Summy nezavisimykh sluchainykh velichin, Nauka, M., 1972 | MR

[13] Rogozin B. A., “Ob uvelichenii rasseivaniya summ nezavisimykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 6:1 (1961), 106–108 | MR

[14] Rudelson M., Vershynin R., “The Littlewood–Offord problem and invertibility of random matrices”, Adv. Math., 218:2 (2008), 600–633 | DOI | MR | Zbl

[15] Rudelson M., Vershynin R., “Smallest singular value of a random rectangular matrix”, Comm. Pure Appl. Math., 62:12 (2009), 1707–1739 | DOI | MR | Zbl

[16] Tao T., Vu Van, “Inverse Littlewood–Offord theorems and the condition number of random discrete matrices”, Ann. of Math., 169:2 (2009), 595–632 | DOI | MR | Zbl

[17] Tao T., Vu Van, “From the Littlewood–Offord problem to the circular law: universality of the spectral distribution of random matrices”, Bull. Amer. Math. Soc. (N. S.), 46:3 (2009), 377–396 | DOI | MR | Zbl