We prove concentration inequalities and associated PAC bounds for continuous- and discrete-time additive functionals for possibly unbounded functions of multivariate, nonreversible diffusion processes. Our analysis relies on an approach via the Poisson equation allowing us to consider a very broad class of subexponentially ergodic processes. These results add to existing concentration inequalities for additive functionals of diffusion processes which have so far been only available for either bounded functions or for unbounded functions of processes from a significantly smaller class. We demonstrate the power of these exponential inequalities by two examples of very different areas. Considering a possibly high-dimensional parametric nonlinear drift model under sparsity constraints, we apply the continuous-time concentration results to validate the restricted eigenvalue condition for Lasso estimation, which is fundamental for the derivation of oracle inequalities. The results for discrete additive functionals are used to investigate the unadjusted Langevin MCMC algorithm for sampling of moderately heavy-tailed densities $π$. In particular, we provide PAC bounds for the sample Monte Carlo estimator of integrals $π(f)$ for polynomially growing functions $f$ that quantify sufficient sample and step sizes for approximation within a prescribed margin with high probability.
Moments and random number generation for the truncated elliptical family of distributions
本文提出了一种从具有严格递减密度母函数的截尾多元椭圆分布族中的任意成员生成随机数的算法。在Neal（2003）和Ho et al.（2012）的基础上，我们通过Gibbs采样器步骤的切片采样算法构建了一种有效的采样方法。我们还提供了一种更快的方法来近似截断多元椭圆分布的一阶和二阶矩，其中蒙特卡罗积分用于截断分区，以及非截断部分的显式表达式（Galarza等人，2020年）。实例和环境空间数据的应用说明了它的实用性。方法在新的R库椭圆中免费提供。
Principal points and elliptical distributions from the multivariate setting to the functional case
随机向量$\mathbf{X}$的$k$主点被定义为使$\mathbf{X}$与集合中最近点之间的期望平方距离最小化的点的集合。Flury（1990，1993）、Tarpey（1995）和Tarpey，Li and Flury（1995）对它们进行了深入研究。对于它们的处理，检查通常限于椭圆分布族。在本文中，我们将以前的结果推广到函数椭圆分布的情况，即，当处理可分希尔伯特空间${\cal H}$上的随机元时。高斯过程的主点在Tarpey和Kinateder（2003）中定义。在本文中，我们推广的概念，主点，自洽点和椭圆分布，使他们适合在这个功能框架。结果连接自洽性和特征向量的协方差算子重新获得在这个新的设置以及一个明确的公式的$k=2$的情况下，以便包括椭圆分布的随机元在${\cal H}$。
On Bernstein Type Inequalities for Stochastic Integrals of Multivariate Point Processes
本文考虑了多元点过程的随机积分，并研究了它们的集中现象。特别地，我们利用Doléans-Dade指数公式得到了一个Bernstein型的浓度不等式，并利用一般的链式变元得到了一个一致指数不等式.作为应用，我们得到了一类泛函指标下离散时间鞅序列的上界，从而得到了非参数极大似然估计的收敛速率，改进了van de Geer的工作.
