Precise moment and tail bounds for Rademacher sums in terms of weak parameters
Let \(r_i\), \(1\leq i \leq n\) be independent symmetric \(\pm\) Bernoulli random variables. Let \(v_i\), \(1\leq i \leq n\) be vectors from a separable Banach space \(F\) associated with the norm \(\|.\|.\) Let \(S= \sum_{k=1}^nv_ir_i.\) It is called a vector-valued Rademacher sum. It is known that, for every \(p>q>0\), there exists some real number \(C\) such that for any \(n\geq 1\) and \(a_i\), \(1\leq i \leq n\) real, \[ \left\|\sum_{k=1}^na_kr_k\right\|_p\leq C\left\|\sum_{k=1}^na_kr_k\right\|_q. \tag{*} \] This inequality is called Khinchin inequality. Let \(K_{p,q}^R\) denote the smallest constant \(C\) for which the inequality (*) holds. It is also known that, for every \(p>q>0\), there exists some real number \(C\) such that for any \(n\geq 1\) and \(v_i,1\leq i \leq n \in F\), \[ \left\|\sum_{k=1}^nv_kr_k\right\|_p\leq C\left\| \sum_{k=1}^nv_kr_k\right\|_q. \tag{**} \] This inequality is called Khinchin-Kahane inequality. Let \(K_{p,q}\) denote the smallest constant \(C\) for which the inequality (**) holds. The author shows that \(\sup_{q\in[1,p]}K_{p,q}/K_{p,q}^R \rightarrow 1\) as \(p \rightarrow \infty\). A random variable \(\theta\) is called a Steinhaus random variable if it has a uniform distribution on the unit circle. The author also obtains tail bounds and estimates for linear combinations of independent Steinhaus random variables.
scientific article; zbMATH DE number 6375691
scientific article