Upper functions for positive random functionals

TL;DR

This paper explores the unifying concept of upper functions for positive random functionals, connecting various phenomena in probability theory and statistics through a common framework.

Contribution

It introduces a novel perspective that links classical probabilistic results and statistical estimation problems via upper functions for positive random functionals.

Findings

Unified approach to probabilistic inequalities and statistical bounds

Connections between law of iterated logarithm and adaptive estimation

Development of upper function techniques for random functionals

Abstract

The main objective of this paper is to look from the unique point of view at some phenomena arising in different areas of probability theory and mathematical statistics. We will try to understand what is common between classical probabilistic results, such as the law of iterated logarithm for example, and well-known problem in adaptive estimation called price to pay for adaptation. Why exists two different kinds of this price? What relates exponential inequalities for M-estimators, so-called uniform-in-bandwidth consistency in density or regression model and the bounds for modulus of continuity of gaussian random functions defined on a metric space equipped with doubling measure? It turned out that all these and many others problems can be reduced to finding upper functions for a collection real- valued random variables. Each variable is the value of a given sub-additive positive functional of continuous random mapping defined on a totally bounded subset of a metric space.