Related papers: Kernel entropy estimation for long memory linear p…

Let $\mathbf {X}=\{X_t, t=1,2,... \}$ be a stationary Gaussian random process, with mean $EX_t=\mu$ and covariance function $\gamma(\tau)=E(X_t-\mu)(X_{t+\tau}-\mu)$. Let $f(\lambda)$ be the corresponding spectral density; a stationary…

We investigate the problem of estimating a function $f$ based on observations from its noisy convolution when the noise exhibits long-range dependence. We construct an adaptive estimator based on the kernel method, derive minimax lower…

Kernel methods represent one of the most powerful tools in machine learning to tackle problems expressed in terms of function values and derivatives due to their capability to represent and model complex relations. While these methods show…

Entropy rate of sequential data-streams naturally quantifies the complexity of the generative process. Thus entropy rate fluctuations could be used as a tool to recognize dynamical perturbations in signal sources, and could potentially be…

We construct in the small-time setting the upper and lower estimates for the transition probability density of a L\'evy process in $\rn$. Our approach relies on the complex analysis technique and the asymptotic analysis of the inverse…

We propose a model selection approach for covariance estimation of a multi-dimensional stochastic process. Under very general assumptions, observing i.i.d replications of the process at fixed observation points, we construct an estimator of…

We provide new asymptotic theory for kernel density estimators, when these are applied to autoregressive processes exhibiting moderate deviations from a unit root. This fills a gap in the existing literature, which has to date considered…

Most machine learning algorithms, such as classification or regression, treat the individual data point as the object of interest. Here we consider extending machine learning algorithms to operate on groups of data points. We suggest…

Loosely speaking, the Shannon entropy rate is used to gauge a stochastic process' intrinsic randomness; the statistical complexity gives the cost of predicting the process. We calculate, for the first time, the entropy rate and statistical…

In many applications of machine learning, a large number of variables are considered. Motivated by machine learning of interacting particle systems, we consider the situation when the number of input variables goes to infinity. First, we…

We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one…

We consider the asymptotic normality in $L^2$ of kernel estimators of the long run covariance kernel of stationary functional time series. Our results are established assuming a weakly dependent Bernoulli shift structure for the underlying…

We discuss algorithms for estimating the Shannon entropy h of finite symbol sequences with long range correlations. In particular, we consider algorithms which estimate h from the code lengths produced by some compression algorithm. Our…

The kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite dimensional Hilbert space. It allows us, for example, to define a distance…

We provide estimates of the rate of strong approximation and bounds for probabilities of moderate deviations in the CLT for the $L_1$-norm of the kernel density estimator without any assumptions on the density and assuming that the kernel…

Matrix approximations are a key element in large-scale algebraic machine learning approaches. The recently proposed method MEKA (Si et al., 2014) effectively employs two common assumptions in Hilbert spaces: the low-rank property of an…

We investigate the computational issues related to the memory size in the estimation of quadratic covariation, taking into account the specifics of financial ultra-high-frequency data. In multivariate price processes, we consider both…

In this paper we consider a linear stochastic Volterra equation which has a stationary solution. We show that when the kernel of the fundamental solution is regularly varying at infinity with a log-convex tail integral, then the…

Large and moderate deviation probabilities play an important role in many applied areas, such as insurance and risk analysis. This paper studies the exact moderate and large deviation asymptotics in non-logarithmic form for linear processes…

This paper addresses a kernel-based learning problem for a network of agents locally observing a latent multidimensional, nonlinear phenomenon in a noisy environment. We propose a learning algorithm that requires only mild a priori…