SCHWEITZER 1996 Problem 10

Let $Y_1 , ..., Y_n$ be exchangeable random variables, ie for all permutations $\pi$ , the distribution of $(Y_{\pi (1)}, \dots, Y_{\pi (n)} )$ is equal to the distribution of $(Y_1 , ..., Y_n)$ . Let $S_0 = 0$ and $$S_j = \sum_{i = 1}^j Y_i j = 1,\dots,n$$ Denote $S_{(0)} , ..., S_{(n)}$ by the ordered statistics formed by the random variables $S_0 , ..., S_n$ . Show that the distribution of $S_{(j)}$ is equal to the distribution of $\max_{0 \le i \le j} S_i + \min_ {0 \le i \le n-j} (S_{j + i} -S_j)$ .