Approximation of probability density functions via location-scale finite mixtures in Lebesgue spaces

Adaptive Bayesian Density Estimation with Location-Scale Mixtures

Abstract: We study convergence rates of Bayesian density estimators based on finite location-scale mixtures of a kernel proportional to exp{−|x|p}. We construct a finite mixture approximation of densities whose logarithm is locally β-Hölder, with squared integrable Hölder constant. Under additional tail and moment conditions, the approximation is minimax for both the Kullback-Leibler divergence...

Fuzzy Best Simultaneous Approximation of a Finite Numbers of Functions

Fuzzy best simultaneous approximation of a finite number of functions is considered. For this purpose, a fuzzy norm on $Cleft (X, Y right )$ and its fuzzy dual space and also the  set of subgradients of a fuzzy norm are introduced. Necessary case of a proved theorem about characterization of simultaneous approximation will be extended to the fuzzy case.

Finite Metric Spaces and Their Embedding into Lebesgue Spaces

The properties of the metric topology on infinite and finite sets are analyzed. We answer whether finite metric spaces hold interest in algebraic topology, and how this result is generalized to pseudometric spaces through the Kolmogorov quotient. Embedding into Lebesgue spaces is analyzed, with special attention for Hilbert spaces, `p, and EN .

Finite Probability Spaces Lecture Notes

ω∈A Pr(ω). In particular, for atomic events we have Pr({ω}) = Pr(ω); and Pr(∅) = 0, Pr(Ω) = 1. The trivial events are those with probability 0 or 1, i. e. ∅ and Ω. The uniform distribution over the sample space Ω is defined by setting Pr(ω) = 1/|Ω| for every ω ∈ Ω. With this distribution, we shall speak of the uniform probability space over Ω. In a uniform space, calculation of probabilities am...

Decomposition of Lebesgue Spaces