International Journal of Mathematics and Mathematical Sciences

Volume 2015 (2015), Article ID 123816, 4 pages

http://dx.doi.org/10.1155/2015/123816

## -Digit Benford Converges to Benford

Department of Science and Mathematics, Columbia College Chicago, Chicago, IL 60605, USA

Received 2 October 2015; Accepted 13 December 2015

Academic Editor: Shyam L. Kalla

Copyright © 2015 Azar Khosravani and Constantin Rasinariu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Using the sum invariance property of Benford random variables, we prove that an -digit Benford variable converges to a Benford variable as approaches infinity.

#### 1. Introduction

Given a positive real number , and a positive integer , we define as the th significant digit of , where and for . Thus, and . We assume base 10 throughout this paper.

Let be the smallest sigma algebra generated by . Then for all and . Within this framework, a random variable is Benford [1–3] if, for all , , and for the probability that the first digits of a real number are is given by

While Benford variables have logarithmic distributions in all of their digits, often times, in Benford literature the focus has only been on the distribution of the first digit. Such limitation may obscure the true nature of the quantity investigated. There are datasets which exhibit a perfect “Benford” distribution in the first digit but fail to do so in the second. Nigrini [4] provided such an example and consequently recommended the use of the first two-digit test in order to improve the recognition of the Benford datasets and thus to identify financial fraud. He also recommended this approach for other accounting related analyses.

Such cases were generalized in [5], where a new class of random variables, called *-digit Benford variables*, was introduced. These variables exhibit a logarithmic digit distribution only in their first digits but are not guaranteed to be logarithmically distributed beyond the th digit. Unlike Benford variables whose decimal logarithm is uniformly distributed mod 1, the decimal logarithm of -digit Benford random variables has less stringent constraints; it must only satisfy prescribed areas over a given partition of the unit interval. This provides us with a collection of random variables that contains the Benford variables as a subset.

It is intuitive to assume that when goes to infinity, an -digit Benford variable converges to Benford. The purpose of this paper is to prove that this is indeed the case.

This paper is structured as follows: in the next section we introduce -digit Benford variables together with some of their properties. In Section 3 we briefly discuss sum invariance, which is fundamental for our main result. Finally, using sum invariance, in Section 4 we show that -digit Benford variable converges to Benford, as .

#### 2. -Digit Benford

An -digit Benford random variable behaves as a Benford variable only in the first -digits but may not have a logarithmic digit distribution beyond th digit [5].

*Definition 1. *Let . A random variable is -digit Benford if for all and all , for ,Note that a Benford variable is -digit Benford variable, for any .

Lemma 2. *If is -digit Benford, then it is -digit Benford, for all .*

*Proof. *Let . Then, by (2)

As an example, let us consider the -digit Benford variable with the probability density function given bywhere . Its graph is illustrated in Figure 1. We can check that . From Lemma 2 this is a 1-digit Benford variable as well. However, is not a -digit Benford variable, since, for example, instead of as required by (2).