Related papers: Nonlinear expectations of random sets
Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In order to establish concentration properties for nonlinear functions $Z(A)$, it is standard to appeal to functional inequalities like Poincar\'e or…
Previsions are positively homogeneous functionals, and are generalized forms of integration functionals. We investigate previsions -- just previsions, not sublinear or superlinear previsions as in previous work. We show that every prevision…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
In this paper we study a family of nonlinear (conditional) expectations that can be understood as a continuous semimartingale with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued…
This paper considers linear rational expectations models in the frequency domain. The paper characterizes existence and uniqueness of solutions to particular as well as generic systems. The set of all solutions to a given system is shown to…
A natural connection between rational functions of several real or complex variables, and subspace collections is explored. A new class of function, superfunctions, are introduced which are the counterpart to functions at the level of…
In recent years, manifold methods have moved into focus as tools for dimension reduction. Assuming that the high-dimensional data actually lie on or close to a low-dimensional nonlinear manifold, these methods have shown convincing results…
Performative prediction is an emerging paradigm in machine learning that addresses scenarios where the model's prediction may induce a shift in the distribution of the data it aims to predict. Current works in this field often rely on…
We consider spline estimates which preserve prescribed piecewise convex properties of the unknown function. A robust version of the penalized likelihood is given and shown to correspond to a variable halfwidth kernel smoother where the…
We consider estimating a compact set from finite data by approximating the support function of that set via sublinear regression. Support functions uniquely characterize a compact set up to closure of convexification, and are sublinear…
We present in this paper the motivation and theory of nonlinear spectral representations, based on convex regularizing functionals. Some comparisons and analogies are drawn to the fields of signal processing, harmonic analysis and sparse…
Functions with uniform level sets can represent orders, preference relations or other binary relations and thus turn out to be a tool for scalarization that can be used, e.g., in multicriteria optimization, decision theory, mathematical…
Law-invariant functionals are central to risk management and assign identical values to random prospects sharing the same distribution under an atomless reference probability measure. This measure is typically assumed fixed. Here, we adopt…
Determining functionals are tools to describe the finite dimensional long-term dynamics of infinite dimensional dynamical systems. There also exist several applications to infinite dimensional {\em random} dynamical systems. In these…
Unravelings provide a probabilistic representation of solutions of master equations and a method of computation of the density operator dynamics. The trajectories generated by unravelings may also be treated as real -- as in the stochastic…
In this paper, we consider the sublinear expectation on bounded random variables. With the notion of uncorrelatedness for random variables under the sublinear expectation, a weak law of large numbers is obtained. With the notion of…
This paper considers the topic of finding prior distributions when a major component of the statistical model depends on a nonlinear function. Using results on how to construct uniform distributions in general metric spaces, we propose a…
In this paper, we obtain a new estimate for uniform integrability under sublinear expectations. Based on this, we establish the limit theorems under nonlinear expectations dominated by sublinear expectations through tightness, and the limit…
Under mild assumptions, we prove that any random multifunction can be represented as the set of minimizers of an infinitely many differentiable normal integrand, which preserves the convexity of the random multifunction. We provide several…
In this paper, we consider a functional linear regression model, where both the covariate and the response variable are functional random variables. We address the problem of optimal nonparametric estimation of the conditional expectation…