In  Heyde proposed an extended version of the well-known Hájek-Rényi inequality to include random variables without finite moments. The purpose of this note is to point out that the theorem in  is in error, and to prove the following theorem in its stead:
Theorem. Let X 1, X 2, ··· be any sequence of random variables, and define be independent random variables, each with mean zero and finite variance, and define Zk = Xk - Yk for k ≧ 1. Let c 1, c 2, ··· be a non-increasing sequence of positive numbers.