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Bivariate partial information decompositions (PIDs) characterize how the information in a "message" random variable is decomposed between two "constituent" random variables in terms of unique, redundant and synergistic information…
Given a process with independent increments $X$ (not necessarily a martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being the Fourier transform of a finite measure $\mu$, we provide explicit Kunita-Watanabe and…
In this work we consider the classical non-linear Boltzmann equation, where the unknown is the distribution function $f$, which depends on the time $t$, the vector $\mathbf{x}$ (the position of a molecule) and its velocity $\mathbf{\xi}$.…
We study the role of randomness in the scrambling of quantum information within integrable free-fermionic systems. Considering quadratic Hamiltonians with varying degrees of randomness, we analyze entanglement-based measures to characterize…
We consider statistics for stochastic evolution equations in Hilbert space with emphasis on stochastic partial differential equations (SPDEs). We observe a solution process under additional measurement errors and want to estimate a real or…
Informative interim adaptations lead to random sample sizes. The random sample size becomes a component of the sufficient statistic and estimation based solely on observed samples or on the likelihood function does not use all available…
In recent years, partially observable functional data has gained significant attention in practical applications and has become the focus of increasing interest in the literature. In this thesis, we build upon the concept of data…
Not all the information in a turbulent field is relevant for understanding particular regions or variables in the flow. Here, we present a method for decomposing a source field into its informative $\boldsymbol{\Phi}_I(\boldsymbol{x},t)$…
We consider the problem of robust deconvolution, and particularly the recovery of an unknown deterministic signal convolved with a known filter and corrupted by additive noise. We present a novel, non-iterative data-driven approach.…
We consider partially observed multiscale diffusion models that are specified up to an unknown vector parameter. We establish for a very general class of test functions that the filter of the original model converges to a filter of reduced…
Common workflows in machine learning and statistics rely on the ability to partition the information in a data set into independent portions. Recent work has shown that this may be possible even when conventional sample splitting is not…
We formulate the statistics of the discrete multicomponent fragmentation event using a methodology borrowed from statistical mechanics. We generate the ensemble of all feasible distributions that can be formed when a single integer…
In this work, the probability uncertainties related to a stationary quantum system with solitonic mass distribution when subjected to deformable hyperbolic potentials are studied. Shannon's entropy and Fisher's information of a…
We consider estimation of the quadratic (co)variation of a semimartingale from discrete observations which are irregularly spaced under high-frequency asymptotics. In the univariate setting, results by Jacod (2008) are generalized to the…
Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix…
We propose a novel estimator for the number of components (denoted by $M$) in a K-variate non-parametric finite mixture model, where the analyst has repeated observations of $K\geq2$ variables that are independent given a finitely supported…
What are the distinct ways in which a set of predictor variables can provide information about a target variable? When does a variable provide unique information, when do variables share redundant information, and when do variables combine…
A simultaneous input and state interval observer is presented for Lipschitz continuous nonlinear systems with unknown inputs and bounded noise signals for the case when the direct feedthrough matrix has full column rank. The observer…
This paper gives a complete characterization of infinitely divisible semimartingales, i.e., semimartingales whose finite dimensional distributions are infinitely divisible. An explicit and essentially unique decomposition of such…
We consider statistical models where functional data are artificially contaminated by independent Wiener processes in order to satisfy privacy constraints. We show that the corrupted observations have a Wiener density which determines the…