Uniform Comparison of Tails of (Non-Symmetric) Probability Measures and Their Symmetrized Counterparts with Applications

Let ($\mathbb{B}$, $\|\cdot\|$) be a separable Banach space and let $\mathcal{M}$ be a class of probability measures on $\mathbb{B}$, and let $\bar{\mu}$ denote the symmetrization of $\mu\in\mathcal{M}$. We provide two sufficient conditions (one in terms of certain quantiles and the other in terms of certain moments of $\|\cdot\|$ relative to μ and $\bar{\mu}$, $\mu\in\mathcal{M}$) for the “uniform comparison” of the μ and $\bar{\mu}$ measure of the complements of the closed balls of $\mathbb{B}$ centered at zero, for every $\mu\in\mathcal{M}$. As a corollary to these “tail comparison inequalities,” we show that three classical results (the Lévy-type Inequalities, the Kwapień-Contraction Inequality, and a part of the Itô–Nisio Theorem) that are valid for the symmetric (but not for the general non-symmetric) independent $\mathbb{B}$-valued random vectors do indeed hold for the independent random vectors whose laws belong to any $\mathcal{M}$ which satisfies one of the two noted conditions and which is closed under convolution. We further point out that these three results (respectively, the tail comparison inequalities) are valid for the centered log-concave, as well as, for the strictly α-stable (or the more general strictly (r, α) -semistable) α ≠ 1 random vectors (respectively, probability measures). We also present several examples which we believe form a valuable part of the paper.