DeepAI AI Chat
Log In Sign Up

Strong laws of large numbers for Fréchet means

by   Steven N. Evans, et al.

For 1 ≤ p < ∞, the Fréchet p-mean of a probability distribution μ on a metric space (X,d) is the set F_p(μ) := min_x∈ X∫_Xd^p(x,y) dμ(y), which is taken to be empty if no minimizer exists. Given a sequence (Y_i)_i ∈ℕ of independent, identically distributed random samples from some probability measure μ on X, the Fréchet p-means of the empirical measures, F_p(1/n∑_i=1^nδ_Y_i) form a sequence of random closed subsets of X. We investigate the senses in which this sequence of random closed sets and related objects converge almost surely as n →∞.


page 1

page 2

page 3

page 4


Strong Laws of Large Numbers for Generalizations of Fréchet Mean Sets

A Fréchet mean of a random variable Y with values in a metric space (𝒬, ...

The intersection of algorithmically random closed sets and effective dimension

In this article, we study several aspects of the intersections of algori...

On the restrictiveness of the hazard rate order

Every element θ=(θ_1,…,θ_n) of the probability n-simplex induces a proba...

One-Pass Graphic Approximation of Integer Sequences

A variety of network modeling problems begin by generating a degree sequ...

The Pareto Record Frontier

For iid d-dimensional observations X^(1), X^(2), ... with independent Ex...

Level sets of depth measures and central dispersion in abstract spaces

The lens depth of a point have been recently extended to general metric ...

A Central Limit Theorem for Martin-Löf Random Numbers

We prove a Central Limit Theorem (CLT) for Martin-Löf Random (MLR) seque...