The geometrical origins of some distributions and the complete concentration of measure phenomenon for mean-values of functionals

TL;DR

This paper explores the geometric origins of various distributions and demonstrates the complete concentration of measure phenomenon for functionals in function spaces, providing new insights into their mean-values and probabilistic behavior.

Contribution

It introduces a geometric derivation of high-order distributions and establishes the measure concentration phenomenon for functionals in p-norm balls of continuous functions.

Findings

Derived high-order normal and exponent distributions geometrically.

Calculated exact mean-values of functionals in function spaces.

Proved the complete measure concentration phenomenon for these functionals.

Abstract

We derive out naturally some important distributions such as high order normal distributions and high order exponent distributions and the Gamma distribution from a geometrical way. Further, we obtain the exact mean-values of integral form functionals in the balls of continuous functions space with $p-$norm, and show the complete concentration of measure phenomenon which means that a functional takes its average on a ball with probability 1, from which we have nonlinear exchange formula of expectation.