An Introduction to Benford's Law by Berger Arno; Hill Theodore P.;

Author:Berger, Arno; Hill, Theodore P.;
Language: eng
Format: epub
Publisher: Princeton University Press
Published: 2015-03-06T16:00:00+00:00

which in turn yields, for every t ≥ 0,

and consequently lim supt→+∞ y(t)/log t ≤ 1. Thus by Proposition 4.8(ii), y is not u.d. mod 1, i.e., x is not Benford.

◼

EXAMPLE 6.63. For the smooth function F (x) = −πx + sin(πx), clearly F (0) = F′(0) = 0, and xF (x) < 0 for all x ≠ 0. By Theorem 6.62, no solution of (6.21) is Benford.

✠

In Theorem 6.62, the requirement that F be C2 can be weakened somewhat [15, Thm. 6.7]. Very similarly to its discrete-time counterpart (Corollary 6.9; cf. also Example 6.11), however, the conclusion may fail if F is merely C1.

EXAMPLE 6.64. For the C1-function F with F (0) = 0 and

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.