Best Constants in Moment Inequalities for Linear Combinations of Independent and Exchangeable Random Variables

In [4] Rosenthal proved the following generalization of Khintchine's inequality: \begin{equation*} \tag{B} \begin{cases} \max \{ \| \sigma^n_{i=1} X_i \|_2, (\sigma^n_{i=1} \| X_i \|^p_p)^{1/p}\} \\ \leq \| \sigma^n_{i=1} X_i \|_p \leq B_p\max \{ \| \sigma^n_{i=1} X_i \|_2, (\sigma^n_{i-i} \| X_i \|^p_p)^{1/p}\} \\ \text{for all independent symmetric random variables} X_1, X_2,\cdots, \text{with finite} pth \text{moment}, 2 < p < \infty.\end{cases}\end{equation*} Rosenthal's proof of (B) as well as later proofs of more general results by Burkholder [1] yielded only exponential of $p$ estimates for the growth rate of $B_p$ as $p \rightarrow \infty$. The main result of this paper is that the actual growth rate of $B_p$ as $p \rightarrow \infty$ is $p/\operatorname{Log} p$, as compared with a growth rate of $\sqrt p$ in Khintchine's inequality.