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Short Communications
A maximal inequality for real numbers with application to exchangeable random variables
S. Levental Michigan State University, Department of Statistics and Probability
Abstract:
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
$$
{\mathbf P}\Bigg\{\max_{1\le j\le 2n}\Bigg|\sum_{i=1}^j X_i\Bigg| > t\Bigg\}\le 16\,{\mathbf P}\Bigg\{\Bigg|\sum_{i=1}^n X_i\Bigg| > \frac{t}{3420}\Bigg\}.
$$
Keywords:
maximal inequality, permutations, exchangeable random variables. Received: 25.08.1999
Language: English
DOI:
10.4213/tvp492