Polynomial approximations of exponential integrals

Exponential Models: Approximations for Probabilities

Welch & Peers (1963) used a root-information prior to obtain posterior probabilities for a scalar parameter exponential model and showed that these Bayes probabilities had the confidence property to second order asymptotically. An important undercurrent of this indicates that the constant information reparameterization provides location model structure, for which the confidence property ...

Gaussian Approximations of Multiple Integrals

Fix k ≥ 1, and let I(l), l ≥ 1, be a sequence of k-dimensional vectors of multiple Wiener-Itô integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as l → +∞ , the law of I(l) is asymptotically close (for example, in the sense of Prokhorov’s distance) to the law of a k-dimensional Gaussian vector having the same covariance matrix as...

Motivic Exponential Integrals

Polynomial First Integrals of Polynomial Differential Systems

In this paper we shall primarily study polynomial integrability of the differential system ẋ = −y + Pn(x, y), ẏ = x + Qn(x, y), n = 2, 3, where Pn and Qn are homogeneous polynomials of degree n. By taking various yet very elementary ways, we not only straightforwardly find the necessary and sufficient integrability conditions but also explicitly present the corresponding polynomial first integr...

Asymptotic approximation of exponential integrals

1 Preliminaries 1 1.1 What this is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Big-O and small-o notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.4 Definition: Asymptotic approximation ...