ACTA issues

Statistical rates of convergence

J. A. Fridy, H. I. Miller, C. Orhan

Acta Sci. Math. (Szeged) 69:1-2(2003), 147-157

Abstract. Most previous rate of convergence comparisons are based on the ordinary limit of the quotient of two null sequences: $x$ converges faster than $z$ provided that $\lim x_{n}/z_{n}=0$. In the present work the quotient limit is weakened to a statistical limit: $x$ converges (stat) faster than $z$ if there is a subset $P\subseteq{\msbm N}$ of natural density one such that $\lim_{n\in P}x_{n}/z_{n}=0$. This study extends results of Bajraktarević and Miller to give conditions on a collection ${\cal A}$ of nonvanishing null sequences that characterize the existence of a nonvanishing null sequence that converges (stat) faster -- or slower -- than each sequence in ${\cal A}$. Other results show when ${\cal A}$ admits a sequence that is statistically completely incomparable to each sequence in ${\cal A}$.

AMS Subject Classification (1991): 40A05, 40C05

Received August 21, 2001. (Registered under 2890/2009.)