Asymptotic properties of rapidly varying functions

Representation and characterization of rapidly varying functions

In this paper representations and characterizations of the class of rapidly varying functions in the sense of de Haan, for index +∞, will be proved. The statements of this theorems will be given in a form that is used by Karamata. Also, some characterization of normalized rapidly varying functions are proved. 2000 AMS Classification: 26A12.

Asymptotic Expansions of Multiple Integrals of Rapidly Oscillating Functions

Expansions of multiple integrals w(aixi,.. .,anxn)g(xi,... ,x„)dxi ■ ■ -dxn, r°i r°n I " Ja\ J a„ where w is a function on R" which is (bk oj,)-periodic in the fcth variable, k = 1,... ,n, and g is smooth, are given in terms of negative powers of the integers o\,.. .,an. Estimates of the remainder term in the expansion are also given. Introduction. Let w be a function with integrable pth power ...

Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

Wederive an asymptotic expansion for the distribution of a compound sumof independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, s...

Asymptotic Behavior of the Nonlinear Schrr Odinger Equation with Rapidly-varying, Mean-zero Dispersion

In this paper we consider the nonlinear Schrr odinger equation with an oscillatory, mean-zero dispersion, which has recently been proposed as an alternative method of dispersion compensation for pulse transmission in optical bers. Under the assumption that the time scale on which the dispersion changes is short in comparison with the dispersion and nonlinearity time scales, we are able to facto...

Asymptotic Behavior of Multivariate Reward Processes with Nonlinear Reward Functions