Asymptotic expansions for probabilities of large deviations
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.
scientific article; zbMATH DE number 3988360