RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription Guidelines for authors Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

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

 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} 

• http://mi.mathnet.ru/eng/tvp4763
• http://mi.mathnet.ru/eng/tvp/v6/i2/p145

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles
Cycle of papers

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