A combinatorial proof of the Burdzy–Pitman conjecture

Abstract

First, we prove the following sharp upper bound for the number of high degree differences in bipartite graphs. Let $(U,V,E)$ be a bipartite graph with $U=\{u_1,u_2,\dots ,u_n\}$ and $V=\{v_1,v_2,\dots ,v_n\}$. For $n\ge k>\frac{n}{2}$ we show that

$$\sum _{1\le i,j\le n}\mathbb{1}\{|\mathrm{deg}(u_i)-\mathrm{deg}(v_j)|\ge k\}\le 2k(n-k).$$

Second, as a corollary, we confirm the Burdzy–Pitman conjecture about the maximal spread of coherent and independent vectors: for $\mathit{\delta }\in (\frac{1}{2},1]$ we prove that

$$\mathbb{P}(|X-Y|\ge \mathit{\delta })\le 2\mathit{\delta }(1-\mathit{\delta })$$

for all random vectors $(X,Y)$ satisfying $X=\mathbb{P}(A|\mathcal{G})$ and $Y=\mathbb{P}(A|\mathcal{H})$ for some event A and independent σ-fields $\mathcal{G}$ and $\mathcal{H}$.