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exchangeable random variables and order statistics

Source: miklos schweitzer 1996 q10

October 11, 2021
probability and stats

Problem Statement

Let Y1,...,YnY_1 , ..., Y_n be exchangeable random variables, ie for all permutations π\pi , the distribution of (Yπ(1),…,Yπ(n))(Y_{\pi (1)}, \dots, Y_{\pi (n)} ) is equal to the distribution of (Y1,...,Yn)(Y_1 , ..., Y_n). Let S0=0S_0 = 0 and Sj=∑i=1jYij=1,…,nS_j = \sum_{i = 1}^j Y_i \qquad j = 1,\dots,n Denote S(0),...,S(n)S_{(0)} , ..., S_{(n)} by the ordered statistics formed by the random variables S0,...,SnS_0 , ..., S_n. Show that the distribution of S(j)S_{(j)} is equal to the distribution of max⁡0≤i≤jSi+min⁡0≤i≤n−j(Sj+i−Sj)\max_{0 \le i \le j} S_i + \min_ {0 \le i \le n-j} (S_{j + i} -S_j).
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