Recurrence relations and Benford’s law

Madeleine Farris (), Noah Luntzlara (), Steven J. Miller (), Lily Shao () and Mengxi Wang ()
Additional contact information
Madeleine Farris: Wellesley College
Noah Luntzlara: University of Michigan
Steven J. Miller: Williams College
Lily Shao: Williams College
Mengxi Wang: University of Michigan

Statistical Methods & Applications, 2021, vol. 30, issue 3, No 3, 797-817

Abstract: Abstract There are now many theoretical explanations for why Benford’s law of digit bias surfaces in so many diverse fields and data sets. After briefly reviewing some of these, we discuss in detail recurrence relations. As these are discrete analogues of differential equations and model a variety of real world phenomena, they provide an important source of systems to test for Benfordness. Previous work showed that fixed depth recurrences with constant coefficients are Benford modulo some technical assumptions which are usually met; we briefly review that theory and then prove some new results extending to the case of linear recurrence relations with non-constant coefficients. We prove that, for certain families of functions f and g, a sequence generated by a recurrence relation of the form $$a_{n+1} = f(n)a_n + g(n)a_{n-1}$$ a n + 1 = f ( n ) a n + g ( n ) a n - 1 is Benford for all initial values. The proof proceeds by parameterizing the coefficients to obtain a recurrence relation of lower degree, and then converting to a new parameter space. From there we show that for suitable choices of f and g where f(n) is nondecreasing and $$g(n)/f(n)^2 \rightarrow 0$$ g ( n ) / f ( n ) 2 → 0 as $$n \rightarrow \infty $$ n → ∞ , the main term dominates and the behavior is equivalent to equidistribution problems previously studied. We also describe the results of generalizing further to higher-degree recurrence relations and multiplicative recurrence relations with non-constant coefficients, as well as the important case when f and g are values of random variables.

Keywords: Benford’s law; Recurrence relations; 11K06; 60F05; 65Q30 (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10260-020-00547-1 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:stmapp:v:30:y:2021:i:3:d:10.1007_s10260-020-00547-1

Ordering information: This journal article can be ordered from
http://www.springer. ... cs/journal/10260/PS2

DOI: 10.1007/s10260-020-00547-1

Access Statistics for this article

Statistical Methods & Applications is currently edited by Tommaso Proietti

More articles in Statistical Methods & Applications from Springer, Società Italiana di Statistica
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().