Necessary and Sufficient Conditions for Unique Solution to Functional Equations of Poincare Type

Necessary and Sufficient Conditions for the Strong Law of Large Numbers for U-statistics

Under some mild regularity on the normalizing sequence, we obtain necessary and sufficient conditions for the Strong Law of Large Numbers for (symmetrized) U-statistics. We also obtain nasc's for the a.s. convergence of series of an analogous form.

Necessary and sufficient conditions of optimal control for infinite dimensional SDEs

A general maximum principle (necessary and sufficient conditions) for an optimal control problem governed by a stochastic differential equation driven by an infinite dimensional martingale is established. The solution of this equation takes its values in a separable Hilbert space and the control domain need not be convex. The result is obtained by using the adjoint backward stochastic differential equation.

关于 Mukhin 局部极限定理有效性充要条件的研究

On Mukhin's necessary and sufficient condition for the validity of the local limit theorem

本文重新检验 Mukhin 于 1984 年提出的局部极限定理充要条件，指出其原始简洁证明无法严格成立，转而证明了一个表述显著不同的较弱充要条件。

On the necessary and sufficient conditions to solve a heat equation with general additive Gaussian noise

In this note we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.

On the necessary and sufficient conditions for Peng's law of large numbers under sublinear expectations

In this paper, we firstly establish the weak laws of large numbers on the canonical space $(\br^\bn,\cb(\br^\bn))$ by traditional truncation method and Chebyshev's inequality as in the classical probability theory. Then we extend them from the canonical space to the general sublinear expectation space. The necessary and sufficient conditions for Peng's law of large numbers are obtained.

A necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk

We consider a branching random walk on the line. Biggins and Kyprianou [6] proved that, in the boundary case, the associated derivative martingale converges almost surly to a finite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In the present paper, we give a necessary and sufficient condition for the non-triviality of this limit and establish a Kesten-Stigum-like result.

A Simple Approach to Functional Inequalities for Non-local Dirichlet Forms

With direct and simple proofs, we establish Poincaré type inequalities (including Poincaré inequalities, weak Poincaré inequalities and super Poincaré inequalities), entropy inequalities and Beckner-type inequalities for non-local Dirichlet forms. The proofs are efficient for non-local Dirichlet forms with general jump kernel, and also work for $L^p (p>1)$ settings. Our results yield a new sufficient condition for fractional Poincaré inequalities, which were recently studied in \cite{MRS,Gre}. To our knowledge this is the first result providing entropy inequalities and Beckner-type inequalitie…

关于空隙律/继承律大数定律的必要和充分条件

Necessary and Sufficient Conditions for the Lacunary/Hereditary Laws of Large Numbers

本文研究了空隙律和继承律大数定律的必要和充分条件，提出更弱的Egorov型条件，并解决了可交换序列中弱大数定律的问题。

The necessary and sufficient conditions in the Marchenko-Pastur theorem

We show that a weak concentration property for quadratic forms of isotropic random vectors ${\bf x}$ is necessary and sufficient for the validity of the Marchenko-Pastur theorem for sample covariance matrices of random vectors having the form $C{\bf x}$, where $C$ is any rectangular matrix with orthonormal rows. We also obtain some general conditions guaranteeing the weak concentration property.

使用广义香农熵刻画有限离散分布的充分必要条件

Necessary and Sufficient Conditions for Characterizing Finite Discrete Distributions with Generalized Shannon's Entropy

本文证明了使用K-1或s-1个正实数阶广义香农熵在置换意义下刻画有限离散分布的充分必要条件，为无序样本空间的推断、拟合优度检验及模型比较提供了标签不变的新基础。

Necessary and sufficient conditions for the existence of the q-optimal measure

This paper presents the general form and essential properties of the q-optimal measure following the approach of Delbaen and Schachermayer (1996) and proves its existence under mild conditions. Most importantly, it states a necessary and sufficient condition for a candidate measure to be the q-optimal measure in the case even of signed measures. Finally, an updated characterization of the q-optimal measure for continuous asset price processes is presented in the light of the counterexample appearing in Cerny and Kallsen (2006) concerning Hobson's (2004) approach.

Necessary and sufficient optimality conditions for relaxed and strict control problems of backward systems

We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex, and the system is governed by a nonlinear backward stochastic differential equation. By introducing a new approach, we establish necessary as well as sufficient conditions of optimality for two models. The first concerns the relaxed controls, who are measure-valued processes. The second is a particular case of the first and relates to strict control problems.