# n-Digit Benford Converges to Benford

Authors: Azar Khosravani; Constantin Rasinariu
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

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## Abstract

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


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## Body

## 1. Introduction

Given a positive real numbery, and a positive integer i, we define Di(y) as the ith significant digit of y, where D1:R+→{1,…,9} and Di:R+→{0,1,…,9} for i>1. Thus, D1(2.718)=2 and D3(2.718)=1. We assume base 10 throughout this paper.LetA be the smallest sigma algebra generated by Di. Then Di-1(d)∈A for all i and d. Within this framework, a random variable Y is Benford [1–3] if, for all m∈N, d1∈{1,…,9}, and di∈{0,1,…,9} for i>1 the probability that the first m digits of a real number are d1d2⋯dm is given by(1)PD1Y=d1,…,DmY=dm=log⁡1+∑j=1m10m-jdj-1.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 n-digit Benford variables, was introduced. These variables exhibit a logarithmic digit distribution only in their first n digits but are not guaranteed to be logarithmically distributed beyond the nth digit. Unlike Benford variables whose decimal logarithm is uniformly distributed mod 1, the decimal logarithm of n-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 whenn goes to infinity, an n-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 introducen-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 n-digit Benford variable converges to Benford, as n→∞.


## 2.n-Digit Benford

Ann-digit Benford random variable behaves as a Benford variable only in the first n-digits but may not have a logarithmic digit distribution beyond nth digit [5].Definition 1.
Letn∈N. A random variable Y is n-digit Benford if for all d1∈{1,…,9} and all di∈{0,1,…,9}, for 2≤i≤n,(2)PD1Y=d1,…,DnY=dn=log⁡1+∑j=1n10n-jdj-1.Note that a Benford variable is n-digit Benford variable, for any n.Lemma 2.
IfY is n-digit Benford, then it is k-digit Benford, for all 1≤k<n.Proof.
Letk=n-1. Then, by (2)(3)PD1Y=d1,…,Dn-1Y=dn-1=∑dn=09PD1Y=d1,…,Dn-1Y=dn-1,DnY=dn=∑dn=09log⁡1+∑j=1n10n-jdj-1=log⁡10n-1d1+⋯+10dn-1+110n-1d1+⋯+10dn-1×⋯×10n-1d1+⋯+10dn-1+1010n-1d1+⋯+10dn-1+9=log⁡10n-1d1+⋯+10dn-1+1010n-1d1+⋯+10dn-1=log⁡1+∑j=1n-110n-jdj-1.As an example, let us consider the2-digit Benford variable Y with the probability density function given by(4)fy=π2yln⁡10sin⁡πβd1d2y,d1+d210≤y<d1+d2+1100,otherwise,where βd1d2(y)=(log⁡10y/10d1+d2)/(log⁡((10d1+d2+1)/(10d1+d2))). Its graph is illustrated in Figure 1. We can check that P(D1(Y)=d1,D2(Y)=d2)=log⁡(1+(10d1+d2)-1). From Lemma 2 this is a 1-digit Benford variable as well. However, Y is not a 3-digit Benford variable, since, for example, P(D1(Y)=1,D2(Y)=1,D3(Y)=1)=2.86×10-3 instead of 3.89×10-3 as required by (2).Figure 1
The pdf of a2-digit Benford variable.


## 3. Sum Invariance

To define sum invariance, we first define the significand function, also known as the mantissa function.Definition 3.
The significand functionS:R+→[1,10) is defined as(5)Sx=10log⁡x-log⁡x,where x denotes the floor of x.Let us consider a finite collection of positive real numbersK and define Sd1⋯dn to be the sum of the significands of the numbers starting with the sequence of digits d1⋯dn. Sum invariance means that Sd1⋯dn is digit independent. For instance, consider the Fibonacci sequence which is known to be Benford [6]. Then for the first 50000 Fibonacci numbers we obtain Table 1, where S1 denotes the sum of all significands starting with 1, and so forth.Table 1
Sum invariance illustration for the first50000 Fibonacci numbers.
S
1
S
2
S
3
S
4
S
5
S
6
S
7
S
8
S
9
21714.0
21712.2
21717.8
21707.4
21713.2
21725.0
21702.7
21717.4
21715.5Nigrini was the first to notice sum invariance in some large collections of data [7]. Allaart [8] 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 n-tuple of leading significant digits is the same as for any other n-tuple: ESd1⋯dnY=ESd1′⋯dn′Y. Allaart showed that a random variable is sum invariant if and only if it is Benford. Berger [3] proved that for sum invariant random variables(6)ESd1⋯dnY=101-nln⁡10.For example, for a Benford sequence with 50000 elements, formula (6) yields S1=⋯=S9=21714.7 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 n-digit Benford variable converges to Benford as n approaches infinity by calculating the bounds for the expected value of its significand.Given a functiong:R→R, we define g†:R→[0,1) as(7)g†x=∑k∈Zgx+k,∀x∈0,1,0,otherwise.Lemma 4.
LetY and X=log⁡Y be two random variables with the probability density functions f and g, respectively. Then(8)ESd1⋯dmY=∫log⁡d1+⋯+dm/10m-1log⁡d1+⋯+dm+1/10m-110xg†xdx.Proof.
Usingf(y)=g(log⁡y)/(yln⁡10), we get (9)ESd1⋯dmY=∫-∞∞Sd1⋯dmyfydy=∑k∈Z∫10kd1+⋯+dm/10m-110kd1+⋯+dm+1/10m-1y10-kglog⁡yyln⁡10dy=∫log⁡d1+⋯+dm/10m-1log⁡d1+⋯+dm+1/10m-110x∑k∈Zgx+kdx.It is known that a necessary and sufficient condition for a random variable to be Benford is thatg†=1 [3, 9]. Consequently, (6) follows immediately from Lemma 4.There are arbitrary many ways in which we can build ann-digit Benford variable. Let Bn be the infinite collection of all n-digit Benford variables. We use ESd1⋯dnBn to denote the collection of the expected values of the significands of the elements of Bn. The next theorem leads to the main result of our paper. It provides the bounds for the expected value ESd1⋯dnY for Y∈Bn.Theorem 5.
LetY∈Bn. Then(10)101-nlog⁡1+1xnxn≤ESd1⋯dnY≤101-nlog⁡1+1xnxn+1,where xn=10n-1d1+⋯+dn.Proof.
We will calculate the lower and upper bounds ofESd1⋯dnY using the fact that ∫0sg†(x)dx is monotonically increasing with s, where g is the probability density function of log⁡Y. From Lemma 4, we obtain(11)ESd1⋯dnY=d1+⋯+dn+110n-1log⁡d1+⋯+dn+110n-1-d1+⋯+dn10n-1log⁡d1+⋯+dn10n-1-∫log⁡d1+⋯+dn/10n-1log⁡d1+⋯+dn+1/10n-110sln⁡10∫0sg†xdxds.Since Y∈Bn, we get(12)∫0sg†xdx=log⁡d1+⋯+dn10n-1+∫log⁡d1+⋯+dn/10n-1sg†xdx.The second term in (12) can take any value between 0 and log⁡(1+1/(10n-1d1+⋯+dn)), since g†(x) is only constrained by its total area over the interval (13)log⁡d1+⋯+dn10n-1,log⁡d1+⋯+dn+110n-1.It follows that(14)101-nlog⁡1+1xnxn≤ESd1⋯dnY,∀Y∈Bn,where xn=10n-1d1+⋯+dn. Similarly we obtain(15)ESd1⋯dnY≤101-nlog⁡1+1xnxn+101-nlog⁡1+1xn,∀Y∈Bn,which completes the proof.Asn→∞, both lower and upper bounds of ESd1⋯dnBn approach 101-n/ln⁡10, proving the sum invariance [3]. Consequently, n-digit Benford variable converges to Benford.



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