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

2017/05/03 by Liu, Cheng-shi
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Probability (math.PR) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1705.01327

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.

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