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### Feldstein, A and Goodman, R (1976)

#### Convergence Estimates for Distribution of Trailing Digits

Journal of the Association for Computing Machinery 23(2), pp. 287-297.

ISSN/ISBN: 0004-5411 DOI: 10.1145/321941.321948

Abstract: This paper analyzes the distribution of trailing digits (tail end digits) of positive real floating-point numbers represented in arbitrary base $\beta$ and randomly chosen from a logarithmic distribution. The analysis shows that the nth digit for $n\ge 2$ is actually approximately uniformly distributed. The approximation depends upon both n and the base $\beta$. It becomes better as n increases, and it is exact in the limit as $n\to \infty$. A table of this distribution is presented for various $\beta$ and n, along with a table of the maximum digit by digit deviation $\Delta$ of the logarithmic distribution from the uniform distribution. Various asymptotic results for $\Delta$ are included. These results have application in resolving open questions of Henrici, of Kaneko and Liu, and of Tsao.

Bibtex:
@article{, author = {Feldstein, Alan and Goodman, Richard}, title = {Convergence Estimates for the Distribution of Trailing Digits}, journal = {J. ACM}, issue_date = {April 1976}, volume = {23}, number = {2}, month = {apr}, year = {1976}, issn = {0004-5411}, pages = {287--297}, url = {https://dl.acm.org/citation.cfm?doid=321941.321948}, doi = {10.1145/321941.321948}, } 

Reference Type: Journal Article

Subject Area(s): Analysis, Numerical Analysis