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: {equation*} {B} {cases} max \{ \| σ^nᵢ₌₁ X_i \|_2, (σ^nᵢ₌₁ \| X_i \|^p_p)1/p\} \\ ≤ \| σ^nᵢ₌₁ X_i \|_p ≤ B_pmax \{ \| σ^nᵢ₌₁ X_i \|_2, (σ^nᵢ₋ᵢ \| X_i \|^p_p)1/p\} \\ for all independent symmetric random variables X_1, X_2,⋯, with finite pth moment, 2 < p < ∞.{cases}{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 Bₚ as p → ∞. The main result of this paper is that the actual growth rate of Bₚ as p → ∞ is p/Log p, as compared with a growth rate of √ p in Khintchine's inequality.