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On the Almost Sure Convergence of Floating‐Point Mantissas and Benford's Law
Author(s) -
Schatte Peter
Publication year - 1988
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19881350108
Subject(s) - mathematics , benford's law , sequence (biology) , law of large numbers , random variable , logarithm , interval (graph theory) , combinatorics , convergence (economics) , discrete mathematics , statistics , mathematical analysis , genetics , economics , biology , economic growth
Let Y 1 , Y 2 ,… be a sequence of random variables and let M n be the floating‐point mantissa of Y n . Further let 1 1, x ) (·) denote the indicator of the interval (1, x ). If Y n / n → Z a.s., where Z ≠ 0 is a further random variable, then the sequence 1 (1, x ) ( M n ) converges a.s. to log x in the sense of ∞‐means and logarithmic means, respectively. The speed of convergence in this relations is estimated. As a conclusion, a further argument for Benford's law is provided.