Related papers: Convergence of Probability Measures and Markov Dec…
We study the convergence of random function iterations for finding an invariant measure of the corresponding Markov operator. We call the problem of finding such an invariant measure the stochastic fixed point problem. This generalizes…
We study optimal solutions to an abstract optimization problem for measures, which is a generalization of classical variational problems in information theory and statistical physics. In the classical problems, information and relative…
Imprecise probability is concerned with uncertainty about which probability distributions to use. It has applications in robust statistics and machine learning. We look at programming language models for imprecise probability. Our…
We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalised convergence proofs along with explicit rates of convergence. More…
Performance metrics (error measures) are vital components of the evaluation frameworks in various fields. The intention of this study was to overview of a variety of performance metrics and approaches to their classification. The main goal…
The idea of slicing divergences has been proven to be successful when comparing two probability measures in various machine learning applications including generative modeling, and consists in computing the expected value of a `base…
The quantum probabilistic convergence in measurement, distinct from mathematical convergence, is derived for indeterminate probabilities from the weak quantum law of large numbers. This is presented in three theorems. The first establishes…
In this paper we give general criteria on tightness and weak convergence of discrete Markov chains to symmetric jump processes on metric measure spaces under mild conditions. As an application, we investigate discrete approximation for a…
In this paper we propose a method to construct probability measures on the space of convex bodies with a given pushforward distribution. Concretely we show that there is a measure on the metric space of centrally symmetric convex bodies,…
We consider the weak convergence of numerical methods for stochastic differential equations (SDEs). Weak convergence is usually expressed in terms of the convergence of expected values of test functions of the trajectories. Here we present…
We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…
In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and possibly conditioned) continuous time…
First, sufficient conditions are given for a triangular array of random vectors such that the sequence of related random step functions converges towards a (not necessarily time homogeneous) diffusion process. These conditions are weaker…
We consider Markov decision processes with synchronizing objectives, which require that a probability mass of $1-\epsilon$ accumulates in a designated set of target states, either once, always, infinitely often, or always from some point…
Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of…
Based on a weak convergence argument, we provide a necessary and sufficient condition that guarantees that a nonnegative local martingale is indeed a martingale. Typically, conditions of this sort are expressed in terms of integrability…
State convergence is essential in several scientific areas, e.g. multi-agent consensus/disagreement, distributed optimization, monotone game theory, multi-agent learning over time-varying networks. This paper is the first on state…
Purpose: A new point of view in the study of impact is introduced. Approach: Using fundamental theorems in real analysis we study the convergence of well-known impact measures. Findings: We show that pointwise convergence is maintained by…
We expand our effective framework for weak convergence of measures on the real line by showing that effective convergence in the Prokhorov metric is equivalent to effective weak convergence. In addition, we establish a framework for the…
A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence…