A central limit theorem for Lebesgue integrals of random fields

Generalized Ito Formulae and Space-Time Lebesgue-Stieltjes Integrals of Local Times

Generalised Ito formulae are proved for time dependent functions of continuous real valued semi-martingales. The conditions involve left space and time first derivatives, with the left space derivative required to have locally bounded 2-dimensional variation. In particular a class of functions with discontinuous first derivative is included. An estimate of Krylov allows further weakening of these conditions when the semi-martingale is a diffusion.

Fatou's Lemma in Its Classic Form and Lebesgue's Convergence Theorems for Varying Measures with Applications to MDPs

The classic Fatou lemma states that the lower limit of a sequence of integrals of functions is greater or equal than the integral of the lower limit. It is known that Fatou's lemma for a sequence of weakly converging measures states a weaker inequality because the integral of the lower limit is replaced with the integral of the lower limit in two parameters, where the second parameter is the argument of the functions. This paper provides sufficient conditions when Fatou's lemma holds in its classic form for a sequence of weakly converging measures. The functions can take both positive and nega…

Calculation of Lebesgue Integrals by Using Uniformly Distributed Sequences in $(0,1)$

We present modified proof of a certain version of Kolmogorov's strong law of large numbers for calculation of Lebesgue Integrals by using uniformly distributed sequences in $(0,1)$. We extend the result of C. Baxa and J. Schoi$β$engeier (cf.\cite{BaxSch2002}, Theorem 1, p. 271) to a maximal set of uniformly distributed (in $(0,1)$) sequences $S_f \subset(0,1)^{\infty}$ which strictly contains the set of sequences of the form $(\{αn\})_{n \in {\bf N}}$ with irrational number $α$ and for which $\ell_1^{\infty}(S_f)=1$, where $\ell_1^{\infty}$ denotes the infinite power of the linear Lebesgue mea…

Representation of Itô Integrals by Lebesgue/Bochner Integrals

In [22], it was proved that as long as the integrand has certain properties, the corresponding Itô integral can be written as a (parameterized) Lebesgue integral (or a Bochner integral). In this paper, we show that such a question can be answered in a more positive and refined way. To do this, we need to characterize the dual of the Banach space of some vector-valued stochastic processes having different integrability with respect to the time variable and the probability measure. The later can be regarded as a variant of the classical Riesz Representation Theorem, and therefore it will be usef…

sinc函数的不当泊松积分

Improper Poisson integral of the sinc function

本文将形式求和视为随机点过程积分，研究其在不可积但存在不当黎曼-勒贝格积分情形下的收敛性，重点分析sinc函数并给出相关关系。

Moment and exponential tail estimations for norms of random variables and random operators in mixed (anisotropic) Lebesgue-Riesz spaces

We study the random variables (r.v.) with values in the so-called mixed (anisotropic) Lebesgue-Riesz spaces: formulate the sufficient conditions for belonging of the r.v. to these spaces, estimate the tail of norms distribution, especially deduce the exponential decreasing tails of them, etc. We obtain as a consequence the estimations of the norms of random integral operators acting between these spaces.

关于Lebesgue积分的一类多元分部积分公式

On a version of a multivariate integration by parts formula for Lebesgue integrals

本文针对可同时不连续、具无穷Hardy-Krause变差的测度诱导函数，提出一类Lebesgue积分的多元分部积分公式，推导了相关性质与收敛结果，可推广至关于连续半连接函数的积分情形。

Uniformly consistently estimating the proportion of false null hypotheses via Lebesgue-Stieltjes integral equations

The proportion of false null hypotheses is a very important quantity in statistical modelling and inference based on the two-component mixture model and its extensions, and in control and estimation of the false discovery rate and false non-discovery rate. Most existing estimators of this proportion threshold p-values, deconvolve the mixture model under constraints on its components, or depend heavily on the location-shift property of distributions. Hence, they usually are not consistent, applicable to non-location-shift distributions, or applicable to discrete statistics or p-values. To elimi…

Lebesgue空间中随机变量的解析构造

An Analytic Construction of Random Variables in Lebesgue Spaces

本文从泛函分析视角发展了Lebesgue空间L^p中随机变量的构造，利用Pettis定理和Riesz表示定理将Bochner积分定义为经典期望的自然推广，扩展了可测性、可积性与期望概念。

Entropy and Grand Lebesgue Spaces approach for Prokhorov-Skorokhod continuity of random processes, with tail estimates

We present in this paper a new sufficient condition for the so-called Prokhorov-Skorokhod continuity of random processes. Our conditions will be formulated in the terms of metric entropy generated by three-dimensional distribution of the considered random process (r.p.) in the parametric set, have a convenient and closed form, and generalize some previous results. We study also the conditions for weak compactness of the sequence of random processes in this space and as a consequence the Central Limit Theorem. Our consideration based on the theory of Prokhorov-Skorokhod spaces of random process…

Matrix Polar Decomposition and Generalisations of the Blaschke-Petkantschin Formula in Integral Geometry

In the work [Bull, Austr. Math. Soc. 85 (2012), 315-234], S.R. Moghadasi has shown how the decomposition of the $N$-fold product of Lebesgue measure on $\mathbb R^n$ implied by matrix polar decomposition can be used to derive the Blaschke-Petkantschin decomposition of measure formula from integral geometry. We use known formulas from random matrix theory to give a simplified derivation of the decomposition of Lebesgue product measure implied by matrix polar decomposition, applying too to the cases of complex and real quaternion entries, and we give corresponding generalisations of the Blaschke…