Towards Empirical Process Theory for Vector-Valued Functions: Metric Entropy of Smooth Function Classes

赋值域中的高斯熵映射

The Gaussian entropy map in valued fields

本文定义了局部域及更一般的非阿基米德赋值域上多元高斯分布的高斯熵映射，证明其像位于超模锥并决定赋值向量分布，且显式计算了3维情形下的像。

Large Deviations of Vector-valued Martingales in 2-Smooth Normed Spaces

We derive exponential bounds on probabilities of large deviations for "light tail" martingales taking values in finite-dimensional normed spaces. Our primary emphasis is on the case where the bounds are dimension-independent or nearly so. We demonstrate that this is the case when the norm on the space can be approximated, within an absolute constant factor, by a norm which is differentiable on the unit sphere with a Lipschitz continuous gradient. We also present various examples of spaces possessing the latter property.

Statistical estimation of the Shannon entropy

The behavior of the Kozachenko - Leonenko estimates for the (differential) Shannon entropy is studied when the number of i.i.d. vector-valued observations tends to infinity. The asymptotic unbiasedness and L^2-consistency of the estimates are established. The conditions employed involve the analogues of the Hardy - Littlewood maximal function. It is shown that the results are valid in particular for the entropy estimation of any nondegenerate Gaussian vector.

Consistency of a nonparametric least squares estimator in integer-valued GARCH models

We consider a nonparametric version of the integer-valued GARCH(1,1) model for time series of counts. The link function in the recursion for the variances is not specified by finite-dimensional parameters, but we impose nonparametric smoothness conditions. We propose a least squares estimator for this function and show that it is consistent with a rate that we conjecture to be nearly optimal.

NPD Entropy: A Non-Parametric Differential Entropy Rate Estimator

The estimation of entropy rates for stationary discrete-valued stochastic processes is a well studied problem in information theory. However, estimating the entropy rate for stationary continuous-valued stochastic processes has not received as much attention. In fact, many current techniques are not able to accurately estimate or characterise the complexity of the differential entropy rate for strongly correlated processes, such as Fractional Gaussian Noise and ARFIMA(0,d,0). To the point that some cannot even detect the trend of the entropy rate, e.g. when it increases/decreases, maximum, or …

带平滑的向量值复合泛函的中心极限定理及其应用

Central limit theorems for vector-valued composite functionals with smoothing and applications

本文研究向量值复合风险泛函，提出高维风险评估框架，并在经验与平滑混合估计量下证明优化复合泛函的新型中心极限定理。