A maximal inequality for real numbers with application to exchangeable random variables

S. Levental

Teoriâ veroâtnostej i ee primeneniâ, Tome 45 (2000) no. 3, pp. 615-621

Résumé

Let $x=(x_1,\ldots, x_n)$ be a sequence of real numbers with ${\sum_{i=1}^n} x_i=0$, and let $\Theta=\{\theta=(\theta_1,\ldots,\theta_n):\theta_i=\pm 1\}$. We will prove that for every $\theta\in\Theta$ and $t\ge 0$ the following holds: $$ \frac{1}{2}\mathbf{P}\{|x_\pi|\ge 38t\}\le \mathbf{P}\{|\theta\cdot x_\pi|\ge t\}\le \mathbf{P}\biggl\{|x_\pi|\ge \frac{t}{2}\biggr\}, $$ where $\mathbf{P}$ stands for the uniform probability on a group $\{\pi\}$ of all permutations of $\{1,\ldots, n\}$, $x_\pi=(x_{\pi(1)},\ldots, x_{\pi(n)})$, $\theta\cdot x_\pi=(\theta_1x_{\pi(1)},\ldots, \theta_nx_{\pi(n)})$, and $|y|=\max_{1\le k\le n}\{|\sum_{i=1}^k y_i|\}$ for every $y=(y_1,\ldots, y_n)\in\mathbf{R}^n$. Our proof is elementary and self-contained. As a corollary of our result we will prove, in the case of real numbers, the following recent result of Pruss [Proc. Amer. Math. Soc., 126 (1998), pp. 1811–1819]: Let $X=(X_1,\ldots, X_{2n})$ be an exchangeable sequence of $2n$ real valued random variables; then for every $t >0$ we have $$ {\mathbf P}\Bigg\{\max_{1\le j\le 2n}\Bigg|\sum_{i=1}^j X_i\Bigg| > t\Bigg\}\le 16\,{\mathbf P}\Bigg\{\Bigg|\sum_{i=1}^n X_i\Bigg| > \frac{t}{3420}\Bigg\}. $$

Détail

Keywords: maximal inequality, exchangeable random variables.
Mots-clés : permutations