Asymptotic expansions for probabilities of large deviations
type: Item
instance of: scholarly article
title: Asymptotic expansions for probabilities of large deviations
zbMATH Open document ID: 0611.60022
DOI: 10.1007/BF00339942
publication date: 1986-01-01T00:00:00Z
review text: Let \(X_ 1,...,X_ n\) be i.i.d. random variables, \(EX_ 1=0\), \(EX^ 2_ 1=1\), \(S_ n=X_ 1+...+X_ n\), \(V_ n(x)=P(S_ n<x)\), \(\Phi\) (x) denotes the standard normal distribution function. Continuing L. Saulis', L. Osipov's, W. Wolf's and his own investigations, the author works with the conditions for the validity of relations of the following type \[ \frac{1-V_ n(x)}{1-\Phi (x/\sqrt{n})}={\mathcal P}_ n(x)\exp [n\sum^{s}_{\nu =3}\mu_{\nu}(x/n)^{\nu}+O(\frac{x^ s}{n^{s- 1}}\epsilon (\frac{n}{x}))],\quad \sqrt{n}\leq x\leq \Lambda (\sqrt{n}),\quad n\to \infty, \] where \(s\geq 2\) is an integer, \({\mathcal P}_ n(x)\) is some polynomial, \(\mu_ 3,...,\mu_ s\) are some real numbers, \(\Lambda\) (z) is such that \(\Lambda\) (z)/z\(\uparrow\), \(\Lambda (z)/z^{1+\epsilon_ 0}\downarrow\), \(0<\epsilon_ 0<1\), \(z\geq z_ 0\), and for \(\epsilon\) (z) the conditions \(\liminf_{z\to \infty} \epsilon (z)=0\), \(\limsup_{z\to \infty} [\epsilon (z)/\epsilon (2z)]<\infty\) are true. In particular, by additional restrictions on the functions \(\Lambda\) (z) and \(\epsilon\) (z) necessary and sufficient conditions are derived for the relation \[ 1-V_ n(x)=[1-\Phi (x/\sqrt{n})]\exp [O(\nu_ s(\Lambda (\sqrt{n})))],\quad n\to \infty, \] to be true uniformly in x in the region \(r\Lambda\) (\(\sqrt{n})\leq x\leq \Lambda (\sqrt{n})\) for any \(r\in (0,1)\) where \[ \nu_ s(z)=\epsilon ((\Lambda^{-1}(z))^ 2/z)z^ s(\Lambda^{-1}(z))^{2s-2}. \] The existence of the moments of \(X_ 1\) beginning with the third order is meant in Cauchy principal value sense.
reviewed by: Tengiz Shervashidze
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 3988360
zbMATH Keywords: asymptotic Cramér-Petrov representation
author: L. V. Rozovsky
MaRDI profile type: Publication
cites work: Q4085018
cites work: Q5582089
cites work: On Probabilities of Large Deviations for Sums of Independent Random Variables
cites work: Q4043852
cites work: Asymptotische Entwicklungen f�r Wahrscheinlichkeiten gro�er Abweichungen
cites work: Q3928731
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 3988360