On the rate of convergence for Takagi class functions

A probabilistic approach to the $Φ$-variation of classical fractal functions with critical roughness

We consider Weierstraß and Takagi-van der Waerden functions with critical degree of roughness. In this case, the functions have vanishing $p^{\text{th}}$ variation for all $p>1$ but are also nowhere differentiable and hence not of bounded variation either. We resolve this apparent puzzle by showing that these functions have finite, nonzero, and linear Wiener--Young $Φ$-variation along the sequence of $b$-adic partitions, where $Φ(x)=x/\sqrt{-\log x}$. For the Weierstraß functions, our proof is based on the martingale central limit theorem (CLT). For the Takagi--van der Waerden functions, we us…

On the $p^{\text{th}}$ variation of a class of fractal functions

The concept of the $p^{\text{th}}$ variation of a continuous function $f$ along a refining sequence of partitions is the key to a pathwise Itô integration theory with integrator $f$. Here, we analyze the $p^{\text{th}}$ variation of a class of fractal functions, containing both the Takagi--van der Waerden and Weierstraß functions. We use a probabilistic argument to show that these functions have linear $p^{\text{th}}$ variation for a parameter $p\ge1$, which can be interpreted as the reciprocal Hurst parameter of the function. It is shown moreover that if functions are constructed from (a skew…

Sharp convergence of nonlinear functionals of a class of Gaussian random fields

We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields. It corresponds to a special case of the general situation considered in [Hairer-Xu], but with improved estimates. As a consequence, we establish convergence of a class of Gaussian fields composite with more general functions. These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.

Convergence of multi-class systems of fixed possibly infinite sizes

Multi-class systems having possibly both finite and infinite classes are investigated under a natural partial exchangeability assumption. It is proved that the conditional law of such a system, given the vector of the empirical measures of its finite classes and directing measures of its infinite ones (given by the de Finetti Theorem), corresponds to sampling independently from each class, without replacement from the finite classes and i.i.d. from the directing measure for the infinite ones. The equivalence between the convergence of multi-exchangeable systems with fixed class sizes and the c…

Flow level convergence and insensitivity for multi-class queueing networks

We consider a multi-class queueing network as a model of packet transfer in a communication network. We define a second stochastic model as a model document transfer in a communication network where the documents transferred have a general distribution. We prove the weak convergence of the multi-class queueing process to the document transfer process. Our convergence result allows the comparison of general document size distributions, and consequently, we prove general insensitivity results for the limit queueing process.

On Obtaining Sharp Bounds of the Rate of Convergence for a Class of Continuous-Time Markov Chains

We study inhomogeneous continuous-time weakly ergodic Markov chains with a finite state space. We introduce the notion of a Markov chain with the regular structure of an infinitesimal matrix and study the sharp upper bounds on the rate of convergence for such class of Markov chains.

Exact convergence rates in the central limit theorem for a class of martingales

We give optimal convergence rates in the central limit theorem for a large class of martingale difference sequences with bounded third moments. The rates depend on the behaviour of the conditional variances and for stationary sequences the rate $n^{-1/2}\log n$ is reached. We give interesting examples of martingales with unbounded increments which belong to the considered class.

一类交换经济的均衡收敛性

Convergence to equilibrium for a class of exchange economies

本文针对“全连接Cobb-Douglas”类交换经济随机动力学模型，证明了其概率分布在全变差度量下指数收敛于唯一均衡，均衡由智能体数量、指数及初始货币商品量决定。

Dynkin类假设下Revuz对应的同胚性

Homeomorphism of the Revuz correspondence under Dynkin class assumptions

研究Dynkin类、局部Dynkin类和Green-tight Dynkin类下，正连续可加泛函与光滑测度之间的Revuz对应的拓扑性质，建立了测度收敛、势收敛和PCAF收敛的双向等价性。

关于一类非线性马尔可夫链收敛速率界的研究

On convergence rate bounds for a class of nonlinear Markov chains

本文提出一种基于谱半径耦合和小扰动思想的新方法，用于评估非线性马尔可夫链的收敛速率，并应用于极值理论的$D$-条件论证。

A weak convergence result for sequential empirical processes under weak dependence

The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying $α$-mixing sequence of random variables. In particular the uniformly boundedness assumption on the function class, which is required in most of the existing literature, is spared. Furthermore under strict stationarity a weak convergence result for the sequential empirical process indexed in function classes is obtained, as a direct consequence. Two examples in mathematical statistics, that cannot be treated with existing results, are given as po…

关于拟无穷可分律的弱收敛性

On weak convergence of quasi-infinitely divisible laws

本文研究了拟无穷可分律的弱收敛准则，在自然假设下将其弱收敛与 Lévy--Khinchine 谱函数的一种特殊收敛联系起来，补充了 Lindner、Pan 和 Sato（2018）的已知结果。

经典紧致群集合的总体收敛速率

Rates of Bulk Convergence for Ensembles of Classical Compact Groups

本文研究了经典紧致群集合在不同群上的随机矩阵收敛速率，发现单位群收敛速率为$ N^{-2} $，正交群和辛群收敛速率为$ N^{-1} $。

Convergence Properties of Kemp's q-Binomial Distribution

We consider Kemp's q-analogue of the binomial distribution. Several convergence results involving the classical binomial, the Heine, the discrete normal, and the Poisson distribution are established. Some of them are q-analogues of classical convergence properties. Besides elementary estimates, we apply Mellin transform asymptotics.

tamed Euler schemes 对于一类随机演变方程的收敛性

Convergence of tamed Euler schemes for a class of stochastic evolution equations

本文研究了一类具有超线性增长算子的随机演变方程的稳定性与收敛性，采用tamed Euler方法结合Galerkin方案进行时空离散化。