On a characterization of probability distribution based on maxima of independent or max-independent random variables

TL;DR

This paper extends a previous result on identifying distributions of independent variables from maxima to a more complex setting involving multiple maxima and known constants, broadening the understanding of distribution characterization.

Contribution

It generalizes Kotlarski's 1978 result to cases with multiple maxima and known coefficients, enhancing the theoretical framework for distribution identification.

Findings

Extended the characterization to complex maxima involving multiple variables.

Proved the identification of distributions under the new maxima structure.

Broadened the theoretical understanding of distribution characterization from maxima.

Abstract

Kotlarski (1978) proved a result on identification of the distributions of independent random variables $X,Y$ and $Z$ from the joint distribution of the bivariate random vector $(U,V)$ where $(U,V)= (\max(X,Z),\max(Y,Z)).$ We extend this result to the case $(U,V)=(\max(X,aZ_1,bZ_2),\max(Y,cZ_1,dZ_2))$ where $X,Y,Z_1,Z_2$ are independent or max-independent random variables, $Z_1$ and $Z_2$ are identically distributed and $a,b,c,d$ are known positive constants.