It has recently been shown that there are substantial differences in the regularity behavior of the empirical process based on scalar diffusions as compared to the classical empirical process, due to the existence of diffusion local time. Besides establishing strong parallels to classical theory such as Ossiander's bracketing CLT and the general Giné-Zinn CLT for uniformly bounded families of functions, we find increased regularity also for multivariate ergodic diffusions, assuming that the invariant measure is finite with Lebesgue density $π$. The effect is diminishing for growing dimension but always present. The fine differences to the classical iid setting are worked out using exponential inequalities for martingales and additive functionals of continuous Markov processes as well as the characterization of the sample path behavior of Gaussian processes by means of the generic chaining bound. To uncover the phenomenon, we study a smoothed version of the empirical diffusion process. It turns out that uniform weak convergence of the smoothed empirical diffusion process under necessary and sufficient conditions can take place with even exponentially small bandwidth in dimension $d=2$, and with strongly undersmoothing bandwidth choice for parameters $β> d/2$ in case $d\geq 3$, assuming that the coordinates of drift and diffusion coefficient belong to some Hölder ball with parameter $β$.
Classical multidimensional scaling is a widely used method in dimensionality reduction and manifold learning. The method takes in a dissimilarity matrix and outputs a low-dimensional configuration matrix based on a spectral decomposition. In this paper, we present three noise models and analyze the resulting configuration matrices, or embeddings. In particular, we show that under each of the three noise models the resulting embedding gives rise to a central limit theorem. We also provide compelling simulations and real data illustrations of these central limit theorems. This perturbation analy…
In this paper, quantitative central limit theorems for $U$-statistics on the $q$-dimensional torus defined in the framework of the two-sample problem for Poisson processes are derived. In particular, the $U$-statistics are built over tight frames defined by wavelets, named toroidal needlets, enjoying excellent localization properties in both harmonic and frequency domains. The Berry-Esséen type bounds associated with the normal approximations for these statistics are obtained by means of the so-called Stein-Malliavin techniques on the Poisson space, as introduced by Peccati, Solé, Taqqu, Utzet…
Adaptive estimator of the memory parameter and goodness-of-fit test using a multidimensional increment ratio statistic
在Surgailis { \it et al.}中首次定义并研究了增量比（IR）统计量。（2007）用于估计平稳或增量平稳高斯过程的记忆参数。在这里，提出了三个扩展的情况下，平稳过程。首先对由多个IR统计量组成的向量建立了多维中心极限定理。其次，可以从该定理推导出拟合优度$χ^2$型检验。最后，这一定理允许构建自适应版本的估计和测试，研究在一般的半参数框架。证明了长记忆参数的自适应估计具有预言性质。模拟证明了有趣的准确性和鲁棒性的估计和测试，即使在非高斯的情况下。
The main objective of this paper is to establish bootstrap uniform functional central limit theorem for Harris recurrent Markov chains over uniformly bounded classes of functions. We show that the result can be generalized also to the unbounded case. To avoid some complicated mixing conditions, we make use of the well-known regeneration properties of Markov chains. We show that in the atomic case the proof of the bootstrap uniform central limit theorem for Markov chains for functions dominated by a function in $L^{2}$ space proposed by Radulović (2004) can be significantly simplified. Finally,…
In this paper we consider some generalizations of the classical d-dimensional Brownian risk model. This contribution derives some non-asymptotic bounds for simultaneous ruin probabilities of interest. In addition, we obtain non-asymptotic bounds also for the case of general trend functions and convolutions of our original risk model.
Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard
我们展示如何里约的方法[概率。Theory Related Fields 104（1996）255- 282]可以适用于在Kantorovich度量的意义下建立某些平稳过程的多维中心极限定理中${\frac{1}{\sqrt{n}}}$的收敛速率。我们给出这个一般结果的两个应用：在克努森气体的情况下，在西奈台球的情况下。
We investigate certain families $X^\hbar$, $0<\hbar \ll 1$, of Gaussian random smooth functions on the $m$-dimensional torus $\mathbb{T}^m_\hbar:=\mathbb{R}^m/(\hbar^{-1}\mathbb{Z} )^m$. We show tha,t for any cube $B\subset \mathbb{R}^m$ of size $<1/2$ and centered at the origin, the number of critical points of $X^\hbar$ in the region $\hbar^{-1}B/(\hbar^{-1}\mathbb{Z} )^m\subset\mathbb{T}^m_\hbar$ has mean $\sim c_1\hbar^{-m}$, variance $\sim c_2\hbar^{-m/2}$, $c_1,c_2>0$, and satisfies a central limit theorem as $\hbar\searrow 0$.
In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and H{ö}lder continuous with exponent $γ$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C \left(1+\mathbf{1}\_{γ=1} \sqrt{\ln(N)}\right)N^{-γ}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio, Gigli and Savar{é} to compu…
We give new contributions on the distribution of the zeros of the Riemann zeta function by using the techniques of the Malliavin calculus. In particular, we obtain the error bound in the multidimensional Selberg' s central limit theorem concerning the zeta zeros on the critical line and we discuss some consequences concerning the asymptotic behavior of the mesoscopic fluctuations of the zeta zeros.
Symmetric heavily tailed random walks on $Z^d, d\geq 1,$ are considered. Under appropriate regularity conditions on the tails of the jump distributions, global (i.e., uniform in $x,t, |x|+t\to\infty,$) asymptotic behavior of the transition probability $p(t,0,x)$ is obtained. The examples indicate that the regularity conditions are essential.
Pathwise uniqueness for multi-dimensional stochastic McKean--Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the solution. For pathwise uniqueness, the drift is assumed to be Dini-continuous in the state variable, while the diffusion must be Lipschitz, continuous in time and uniformly nondegenerate. The setting is classical McKean--Vlasov, that is, coefficients of the equation are represented as integrals over the marginal distributions of the process.
The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero th…
We study the quasi-stationary behavior of multidimensional processes absorbed when one of the coordinates vanishes. Our results cover competitive or weakly cooperative Lotka-Volterra birth and death processes and Feller diffusions with competitive Lotka-Volterra interaction. To this aim, we develop original non-linear Lyapunov criteria involving two Lyapunov functions, which apply to general Markov processes.
Rio gave a concise bound for the central limit theorem in the Vaserstein distances, which is a ratio between some higher moments and some powers of the variance. As a corollary, it gives an estimate for the normal approximation of the small jumps of the Lévy processes, and Fournier applied that to the Euler approximation of Lévy-driven stochastic differential equations. However both results are restricted to the one-dimensional case. It will be shown in this article that, following Davie's idea, one can generalise Rio's result to multidimensional case, and the number of moments required is ind…
We give a new, self-contained proof of the multidimensional central limit theorem using the technique of ``doubling variables," which is traditionally used to prove uniqueness of solutions of partial differential equations (PDEs). Our technique also yields quantitative bounds for random variables with finite $2+γ$ moment for some $γ\in (0,1]$; when $γ=1$, this proves a version of the Berry--Esseen theorem in $\mathbb{R}^d$.
Small deviations in p-variation norm for multidimensional Levy processes
设Z是一个具有强有限p-变差的Rd-值Levy过程，其中p<2。证明了由Z通过零化广义漂移得到的“非补偿"过程Y在p-变差下具有小偏差性质。这个性质意味着零函数属于关于p-变化距离的Y定律的支持。由于T. J. Lyons/D. R. E. Williams，这使我们能够证明一个关于由Z驱动的正则向量的p-Skorohod距离的支撑定理，而无需对Z作任何假设，改进了H.国田我们还给出了一个准则，确保小偏差性质Z本身，注意到，下的均匀距离，我们已经得到了在以前的文件，不再持有下的p-变化距离的特征。
