^{1}

^{2}

^{1}

^{2}

Fix a base

Benford's law gives the expected frequencies of the
digits in many tabulated data. It was first observed by Newcomb in the 1880s,
who noticed that pages of numbers starting with a

For any base

We can prove many mathematical systems
follow Benford's law, ranging from recurrence
relations [

This work is motivated by two observations (see Remark

Proving our results requires analyzing the
distribution of digits of independent random variables drawn from the standard
exponential, and quantifying how close the distribution of digits of a random
variable with the standard exponential distribution is to Benford's law.
Leemis et al. [

Both proofs apply Fourier analysis to periodic
functions. In [

A sequence

A positive sequence (or values of a function) is
Benford base

We use the following notations for the various error terms.

Let

Big-Oh notation: for

The following theorem is the starting point for investigating the distribution of digits of order statistics.

Let

The above theorem was proved in [

Let

We briefly describe the reasons behind this notation.
One important property of Benford's law is that it is invariant under
rescaling; many authors have used this property to characterize Benford
behavior. Thus, if a dataset is Benford base

The situation is different for exponential behavior.
Multiplying all elements by a fixed constant

We consider a simple case first, and show how the more
general case follows. Let

For uniformly distributed random variables, if we know
the distribution of

As

Let

A similar result holds for other
distributions.

Let

The key ingredient in this generalization is that the techniques, which show that the differences between uniformly distributed random variables become independent exponentially distributed random variables, can be modified to handle more general distributions.

We restricted ourselves to a subset of all consecutive
spacings because the normalization factor changes throughout the domain. The
shift in the shifted exponential behavior depends on which set of

Let

The conditions
of Theorem

All 499 999 differences of adjacent order statistics
from 500 000 independent random variables from the Pareto distribution with
minimum value and variance

The situation is very different if instead we study
normalized differences:

Assume the
probability distribution

Appropriately
scaled, the distribution of the digits of the differences is universal, and is
the exponential behavior of Theorem

The main
motivation for this work is the need for improved ways of assessing the
authenticity and integrity of scientific and corporate data. Benford's law has
been successfully applied to detecting income tax, corporate, and voter fraud
(see [

The paper is organized as follows. We prove Theorem

Theorem

To prove Theorem

For each

We assume that in the interval

We now investigate the order statistics of the

We have an interval of size

Some care is required in these calculations. We have a
conditional probability as we assume that

Recalling our expansion for

The calculation of the second probability, the
conditional probability that the

We generalize the notation from Section

Before we
considered just one fixed interval; as we are studying

Similar to (

Similar to (

Let

We now prove our theorems which determine when these
bin-dependent shifts cancel (yielding Benford behavior), or reinforce (yielding
sums of shifted exponential behavior).

There are
approximately

Thus, each of the

Thus, the bin-dependent shift in the shifted
exponential behavior is approximately

Consider the
case when the density is a uniform distribution on some interval. Then, all

We analyze the assumptions of Theorem

If we choose any distribution

If

To see this, note that the cumulative distribution
function of

The condition from (

Let

We chose to
study a Pareto distribution because the distribution of digits of a random
variable drawn from a Pareto distribution converges to Benford behavior (base
10) as

Modifying the proof of Theorem

If

Thus, our assumptions on

As an example of Theorem

The universal
behavior of Theorem

To prove Theorem

As Benford behavior is equivalent to

We use the fact that the derivative of the infinite
sum

Let

The above
series expansion is rapidly convergent, and shows the deviations of

We can improve (

With constants as in the theorem, if we take

Let

Consider
the infinite series expansion in (

We show why in addition to

Let

For each

The authors would like to thank Ted Hill, Christoph Leuenberger, Daniel Stone, and the referees for numerous helpful comments. S. J. Miller was partially supported by NSF (Grant no. DMS-0600848).