Statistically pre-Cauchy sequences and bounded moduli

Abstract

Let x=(x_k) be a sequence and let f be a bounded modulus. We prove that x is statistically pre-Cauchy if and only if 

lim_n 1/n²∑_j,k≤nf(|x_k−x_j|)=0.

This implies a theorem due to Connor, Fridy and Kline [4].