On the best constant in Marcinkiewicz–Zygmund inequality

Abstract

Let { X n , n⩾1} be a sequence of independent r.v.'s with EX n=0, C(p) be the best constant in the following Marcinkiewicz–Zygmund inequality: E ∑ k=1 n X k p⩽C(p)n p/2−1 ∑ k=1 n E|X k| p, p⩾2. In this paper we prove that [ C( p)] 1/ p grows like p as p→∞ and give an estimate C(p)⩽(3 2 ) pp p/2 .