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Asymptotic expansions for probabilities of large deviations. (English) Zbl 0611.60022

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.
Reviewer: T.Shervashidze

MSC:

60F10 Large deviations
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[1] Petrov, V., Sums of independent random variables (1975), Berlin: Akademie-Verlag, Berlin · Zbl 0322.60043
[2] Saulis, L., An asymptotic expansion for probabilities of large deviations for sums of independent random variables (Russian), Litovsk. Mat. Sb., 9, 605-625 (1969) · Zbl 0188.23701
[3] Osipov, L., On probabilities of large deviations for sums of independet random variables (Russian), Teor. Verojatnost. i. Primenen., 17, 320-341 (1972) · Zbl 0271.60035
[4] Saulis, L., The limit theorems taking in account the large deviations under Ju.V. Linnik’s condition (Russian), Litovsk. Mat. Sb., 13, 173-196 (1973) · Zbl 0292.60045
[5] Wolf, W., Asymptotische Entwicklungen für Wahrscheinlichkeiten großer Abweichungen. 2. Wahrscheinlichkeitstheor, Verw. Geb., 40, 239-256 (1977) · Zbl 0365.60033
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