I read Wassily Hoeffding's paper "a class of statistics with asymptotically normal distribution". In proving "$n\sigma^{2}(U_{n})$ is decreasing in n" in Theorem 5.2, it simply says "using (5.33) and (5.31)". Yet this is not obvious to me. Is there any other proof I can find to read? Thank you.
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The claim is that $\sum_{c=1}^m cd_{n,c}=0$. The formula (5.33) tells you exactly what $d_{n,c}$ is: it is the difference of two products involving binomial coefficients. When you multiply by $c$ and sum over $c$, each product produces a sum that is just like the left hand side of (5.31). By (5.31), $$\sum_{c=1}^m cd_{n,c}= n\cdot \frac{m^2}{n} - (n+1)\cdot \frac{m^2}{n+1} = 0$$