Convergence Estimates for the Distribution of Trailing Digits

Abstract

This paper analyzes the distribution of trailing digits (tail end digits) of positive real floating-point numbers represented in arbitrary base β and randomly chosen from a logarithmic distribution. The analysis shows that the nth digit for n ≥ 2 is actually approximately uniformly distributed. The approximation depends upon both n and the baseβ. It becomes better as n increases, and it is exact in the limit as n ⇒ ∞. A table of this distribution is presented for various β and n, along with a table of the maximum digit by digit deviation Δ of the logarithmic distribution from the uniform distribution. Various asymptotic results for Δ are included. These results have application in resolving open questions of Henrici, of Kaneko and Liu, and of Tsao. © 1976, ACM. All rights reserved.