Special cases of second order Wiener germ approximations (Q909326)

The paper is concerned with the distribution functions of densities of the form \[ f_{\epsilon}(x)\sim const.(2\pi \epsilon)^{-1/2}\exp \{- K(x)/\epsilon \}{\mathcal D}(x)\exp \{\epsilon S(x)\} \] where the functions K(x), S(x), \({\mathcal D}(x)\) satisfy certain conditions. Such family of densities, roughly speeking, is called Wiener germ, a mathematical theory which the author has initiated. Some results of this theory are applied to get approximations for the tail probabilities of various interesting distributions (e.g. Student's t-distribution, beta-distribution, gamma- distribution etc.). Sums of i.i.d. random variables are investigated, too.