Asymptotic results for weighted means of linear combinations of independent Poisson random variables

ABSTRACT

In this paper we prove the large deviation principle for a class of weighted means of linear combinations of independent Poisson distributed random variables, which converge weakly to a normal distribution. The interest in these linear combinations is motivated by the diffusion approximation in Lansky [On approximations of Stein's neuronal model, J. Theoret. Biol. 107 (1984), pp. 631–647] of the Stein's neuronal model (see Stein [A theoretical analysis of neuronal variability, Biophys. J. 5 (1965), pp. 173–194]). We also prove an analogue result for sequences of multivariate random variables based on the diffusion approximation in Tamborrino, Sacerdote, and Jacobsen [Weak convergence of marked point processes generated by crossings of multivariate jump processes. Applications to neural network modeling, Phys. D 288 (2014), pp. 45–52]. The weighted means studied in this paper generalize the logarithmic means. We also investigate moderate deviations.     

Some Useful Integral Representations for Information-Theoretic Analyses

This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact formulas for quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive non-integer real (also known as fractional moments). The proposed formula is applied to a variety of examples with an information-theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, n, of nonnegative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over n dimensions, as needed in the direct calculation of the desired moment.     

Norms of embeddings of logarithmic Bessel potential spaces

Let be a subset of with finite volume, let and let be a Young function with for large . We show that the norm on the Orlicz space is equivalent to

We also obtain estimates of the norms of the embeddings of certain logarithmic Bessel potential spaces in which are sharp in their dependences on provided that is large enough.

     

一类函数的对数完全单调性

通过对参数λ,μ的讨论,主要利用函数的单调性理论,已有对数完全单调函数的性质以及幂函数的积分表达式研究了函数Gλ,μ(x)及函数[Gλ,μ(x)]-1的对数完全单调性,并在此基础上得到了一定条件下函数Gλ,μ(x)及[Gλ,μ(x)]-1对数完全单调的充要条件.     

我国海运出口贸易碳排放影响因素的对数指数分解研究

对2005-2013年我国的碳排放进行了对数指数分解,得出海运出口贸易规模、能源效率、能源结构、碳排放强度是构成碳排放的四个因素,其中海运出口规模和碳排放强度是碳排放的正向拉动关系,能源结构和能源效率是碳排放的负向抑制关系.运用协整检验证明四个因素与碳排放之间具有长期稳定的关系,运用Granger因果检验证明海运出口贸易规模、能源结构、能源效率是碳排放的Granger原因.在此基础上,提出减少海运业碳排放的政策建议.     

A sharp inequality for the logarithmic coefficients of univalent functions

We prove the sharp inequality

for the logarithmic coefficients of a normalized univalent function in the unit disk.

     

Full collocation methods for some boundary integral equations

In this paper we propose a fully discretized version of the collocation method applied to integral equations of the first kind with logarithmic kernel. After a stability and convergence analysis is given, we prove the existence of an asymptotic expansion of the error, which justifies the use of Richardson extrapolation. We further show how these expansions can be translated to a new expansion of potentials calculated with the numerical solution of a boundary integral equation such as those treated before. Some numerical experiments, confirming our theoretical results, are given. This revised version was published online in August 2006 with corrections to the Cover Date.