Rearrangements of distributions on integers that minimize variance
Aistis Atminas, Valentas Kurauskas
TL;DR
This work establishes a discrete analogue of rearrangement inequalities by showing that, for any finite integer-supported distribution, the variance is minimized by a specific left-right symmetric rearrangement $X^+$ that assigns the largest probability to $0$, the next to $1$, then to $-1$, and so on. The authors place the result in a general dispersion framework $D_f(X)$ with minimizers $M_f(X)$ and leverage a rearrangement-based comparison to prove $\mathrm{Var}\, X^+ \le \mathrm{Var}\, X$, with equality only in translations or sign-flips of the distribution. The main proof analyzes $D_f(X)$ via a sorted probability vector and the corresponding distance-function vectors, using the distance to the nearest integer $a'$ and a constructed nondecreasing vector ${\bf w}$ to bound $D_f(X)$ by $D_f(X^+)$. The results specialize to $f(x)=x^2$ (variance) and related $f$ yielding MAD, providing a concise, self-contained discrete analogue to continuous rearrangement phenomena and informing equality cases in discrete inverse problems.
Abstract
Which permutations of a probability distribution on integers minimize variance? Let $X$ be a random variable on a set of integers $\{x_1, \dots, x_N\}$ such that $\mathbb{P}(X_i = x_i) = p_i$, $i \in \{1,\dots,N\}$. Let $(p^{(1)}, \dots, p^{(N)})$ be the sequence $(p_1, \dots, p_N)$ ordered non-increasingly. Let $X^+$ be the random variable defined by $\mathbb{P}(X=0)=p^{(1)}$, $\mathbb{P}(X=1) = p^{(2)}$, $\mathbb{P}(X=-1)=p^{(3)}, \dots, \mathbb{P}(X=(-1)^N \lfloor \frac {N} 2 \rfloor)=p^{(N)}$. In this short note we generalize and prove the inequality $\mathrm{Var}\, X^+ \le \mathrm{Var}\, X$.