Research Article | Open Access
-Digit Benford Converges to Benford
Using the sum invariance property of Benford random variables, we prove that an -digit Benford variable converges to a Benford variable as approaches infinity.
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  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 , 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 .
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).
3. Sum Invariance
To define sum invariance, we first define the significand function, also known as the mantissa function.
Definition 3. The significand function is defined aswhere denotes the floor of .
Let us consider a finite collection of positive real numbers and define to be the sum of the significands of the numbers starting with the sequence of digits . Sum invariance means that is digit independent. For instance, consider the Fibonacci sequence which is known to be Benford . Then for the first Fibonacci numbers we obtain Table 1, where denotes the sum of all significands starting with , and so forth.
Nigrini was the first to notice sum invariance in some large collections of data . Allaart  refined this concept, by defining it in connection with continuous random variables. Specifically, a distribution is sum invariant if the expected value of the significands of all entries starting with a fixed -tuple of leading significant digits is the same as for any other -tuple: . Allaart showed that a random variable is sum invariant if and only if it is Benford. Berger  proved that for sum invariant random variablesFor example, for a Benford sequence with elements, formula (6) yields rounded to the tenths, which is very close to the actual values for the Fibonacci numbers illustrated in Table 1. Naturally, the more the numbers are taken from the sequence, the closer the one gets to the theoretical sum.
4. Main Result
A random variable is sum invariant if and only if it is Benford [3, 8]. Using this result, we will prove that an -digit Benford variable converges to Benford as approaches infinity by calculating the bounds for the expected value of its significand.
Given a function , we define as
Lemma 4. Let and be two random variables with the probability density functions and , respectively. Then
Proof. Using , we get
There are arbitrary many ways in which we can build an -digit Benford variable. Let be the infinite collection of all -digit Benford variables. We use to denote the collection of the expected values of the significands of the elements of . The next theorem leads to the main result of our paper. It provides the bounds for the expected value for .
Theorem 5. Let . Thenwhere .
Proof. We will calculate the lower and upper bounds of using the fact that is monotonically increasing with , where is the probability density function of . From Lemma 4, we obtainSince , we getThe second term in (12) can take any value between and , since is only constrained by its total area over the interval It follows thatwhere . Similarly we obtainwhich completes the proof.
As , both lower and upper bounds of approach , proving the sum invariance . Consequently, -digit Benford variable converges to Benford.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
- F. Benford, “The law of anomalous numbers,” Proceedings of the American Philosophical Society, vol. 78, no. 4, pp. 551–572, 1938.
- T. P. Hill, “Base-invariance implies Benford's law,” Proceedings of the American Mathematical Society, vol. 123, no. 3, pp. 887–895, 1995.
- A. Berger and T. P. Hill, “A basic theory of Benford's law,” Probability Surveys, vol. 8, pp. 1–126, 2011.
- M. J. Nigrini, Benford's Law: Applications for Forensic Accounting, Auditing, and Fraud Detection, John Wiley & Sons, Hoboken, NJ, USA, 2012.
- A. Khosravani and C. Rasinariu, “n-digit benford distributed random variables,” Advances and Applications in Statistics, vol. 36, no. 2, pp. 119–130, 2013.
- R. L. Duncan, “An application of uniform distribution to the Fibonacci numbers,” The Fibonacci Quarterly, vol. 5, pp. 137–140, 1967.
- M. J. Nigrini, The detection of income tax evasion through an analysis of digital frequencies [Ph.D. thesis], University of Cincinnati, Cincinnati, Ohio, USA, 1992.
- P. C. Allaart, “An invariant-sum characterization of Benford's law,” Journal of Applied Probability, vol. 34, no. 1, pp. 288–291, 1997.
- P. Diaconis, “The distribution of leading digits and uniform distribution mod 1,” The Annals of Probability, vol. 5, no. 1, pp. 72–81, 1977.
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.