On Exact Maximal Khinchine Inequalities

Abstract

Let ε ₁ …, ε _n be independent Rademacher random variables, and let a _l, …, a _n be real numbers such that $a_1^2 + \ldots a_n^2 = 1$. Let (B _t ) be a standard Brownian motion. One has the generalized moment identity

$$Eg\left( {\mathop {0 \leqslant t \leqslant 1|x + {B_t}|}\limits^{\max } } \right) = 2{\sum\limits_{n = 1}^\infty {\left( {-1} \right)} ^{n-1}}Eg\left( {x \vee \frac{{|x + {B_1}|}}{{2n-1}}} \right),$$

where x > 0 and g is non-decreasing with g(x) = 0 and Eg(|x + B ₁|) < ∞. This identity is used to show that the inequality

$${\rm E}\mathop {\max }\limits_{1 \leqslant k \leqslant n} \left| {\sum\limits_{i = 1}^k {a_i \varepsilon _i } } \right|^p \leqslant {\rm E}\mathop {\max }\limits_{0 \leqslant t \leqslant 1} \left| {B_t } \right|^p ,$$

conjectured in [5], fails in a ne’ghborhood of p = 1. However, this inequality is proved to hold for p ≥ 3 in the case a _l=⋯=a _a=1/$\sqrt {n;}$ the latter result is based on a “discrete” version of the above generalized moment identity.