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 Teor. Veroyatnost. i Primenen., 1961, Volume 6, Issue 2, Pages 145–163 (Mi tvp4763) Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations. I

Yu. V. Linnik

Moscow

Abstract: The independent identically distributed variables $x_1,x_2,…,x_n$ are supposed to have $E({x_j})=0$; $D({x_j})=\sigma^2<\infty$. Denote
$$Z_n=\frac{x_1+\cdots+x_n}{\sigma\sqrt n}.$$
Let $\Psi(n)\to\infty$ be some monotone function. The sequence of segments $[0,\Psi (n)]$ is called the zone of normal attraction (z. n. a.) if
$$\frac{{\mathbf P(Z_n>x)}}{\frac1{\sqrt{2\pi}}\int_x^\infty{e^{-n^2/2} dn}}\to1$$
for $x\in[0,\Psi(n)]$; the zones $[-\Psi(n),0]$ are defined similarly as z. n. a. The zones $[0,n^\alpha];[-n^\alpha,0](\alpha>0$ constant) are called simplest. The zones such that $\Psi(n)=o(n^{1/6})$ are called “narrow”.
For the random variables of the class $(d)$ (possessing a bounded continuous density) the zones $[0,\Psi (n)],[-\Psi (n),0]$ are called the zones of the uniform local normal attraction (z. u. l. n. a.) if
$$\frac{p_{Z_n}(x)}{\frac1{\sqrt{2\pi}}e^{-x^2/2}}\to1$$
uniformly in x belonging to the said zones. Let $\alpha<1/2$. The condition
$$\mathbf E\exp|{x_j}|^{4\alpha/(2\alpha+1)}<\infty$$
is proved to be necessary for the zones $[0,n^\alpha],[-n^\alpha,0]$, to be z. n. a., and for $x_j\in(d)$ to be the z. u. l. n. a. Let $\rho(n)$ be a given monotonic function increasing as slowly as we please, then the condition $(*)$ is sufficient for the zones $[0,n^\alpha/\rho(n)];[-n^\alpha/\rho(n),0]$ to be the z. n. a., and for $x_j\in(d)$ to be the z. u. l. n. a. if $\alpha<1/6$. If $\alpha>1/6$, $x_j\in(d)$, a condition is given in terms of the series $1/6,1/4,3/10,…,(1/2)(s+1)/(s+3)\to1/2$ and of moments of $x_j$. This condition is necessary for the zones $[0,n^\alpha \rho (n)]$, $[-n^\alpha\rho(n),0]$ to be z. u. l. n. a. and sufficient for the zones $[0,n^\alpha/\rho (n)]$; $[-n^\alpha\rho (n),0]$ to be z. u. l. n. a. Full text: PDF file (1499 kB)

English version:
Theory of Probability and its Applications, 1961, 6:2, 131–148 Citation: Yu. V. Linnik, “Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations. I”, Teor. Veroyatnost. i Primenen., 6:2 (1961), 145–163; Theory Probab. Appl., 6:2 (1961), 131–148 Citation in format AMSBIB
\Bibitem{Lin61} \by Yu.~V.~Linnik \paper Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations.~I \jour Teor. Veroyatnost. i Primenen. \yr 1961 \vol 6 \issue 2 \pages 145--163 \mathnet{http://mi.mathnet.ru/tvp4763} \transl \jour Theory Probab. Appl. \yr 1961 \vol 6 \issue 2 \pages 131--148 \crossref{https://doi.org/10.1137/1106019} 

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This publication is cited in the following articles:
1. Theory Probab. Appl., 48:3 (2004), 528–535      2. D. V. Batkovich, “Local limit theorems for large deviations”, J. Math. Sci. (N. Y.), 188:6 (2013), 641–654   3. Lifshits M.A. Nikitin Ya.Yu. Petrov V.V. Zaitsev A.Yu. Zinger A.A., “Toward the History of the Saint Petersburg School of Probability and Statistics. i. Limit Theorems For Sums of Independent Random Variables”, Vestnik St. Petersburg Univ. Math., 51:2 (2018), 144–163  •   Contact us: math-net2021_12 [at] mi-ras ru Terms of Use Registration to the website Logotypes © Steklov Mathematical Institute RAS, 2021