FFourier Analysis and Benford Random Variables, Page 1
Fourier Analysis and Benford Random Variables
Frank BenfordJune, 2020
1. Introduction
To forestall potential confusion, let me say at the outset that the author of this paper isnot the physicist Frank Benford for whom Benford's law is named, but one of hisgrandsons.This paper has several major purposes. Perhaps the central purpose is to summarizesome results from investigations into base dependence of Benford random variables,including a discussion of Whittaker's astonishing random variable. The principal tools usedto derive these results are Fourier series and Fourier transforms, and a second majorpurpose of this paper is to present an introductory exposition about these tools. Since thereexist many excellent introductions to Fourier series, and a fair number of undergraduatelevel introductions to Fourier transforms, one may wonder if another introduction is reallyneeded. My motivation for writing this paper is twofold. First, I believe that the theory ofBenford random variables and the “Benford analysis” of a positive random variable hassome interest and deserves to be better known. Second, I think Benford analysis provides areally excellent illustration of the utility of Fourier series and transforms, and revealscertain interconnections between series and transforms that are not obvious from the usualway these subjects are introduced.
A note to student readers . This paper in neither a research paper nor a survey of theliterature. Instead, it's intended as an undergraduate level introduction to both Benfordrandom variables and Fourier analysis. Consequently, the reader may have an imperfectunderstanding of some of the language. For example, you may not understand every detailin the statement “The space is a Hilbert space.” If you don't understand everything in this sentence, my advice is not to worry about it. You should be able to get the gist ofthe ideas presented here without having a deep understanding of all of the language. If thelanguage piques your interest, ask one of your professors (who may suggest that you'll wantto enroll in a course in real analysis.)
2. Benford Random Variables
My grandfather [1] studied the empirical distribution of “first digits” (i.e., the leftmostdigit, or most significant digit) of randomly selected numbers by compiling a large (for1938) and heterogeneous data set of 20,229 (decimal) numbers culled from newspapers,almanacs, scientific handbooks, and other sources. He found that the proportion of these numbers with first digit (for every ) was given to a close approximation by the formula This was actually a rediscovery, as Simon Newcomb had made the same observation in 1881. ourier Analysis and Benford Random Variables, Page 2Pr log (2.1)where , for the moment, denotes “proportion.” This formula is “The First Digit Law.”Pr The observed and expected proportions of each of the nine possible first digits (fromgrandfather's data set and eq. (2.1), respectively) are shown in the following figure.
The First Digit Law
Observed Expected
No finite data set can satisfy eq. (2.1) exactly, as the proportion of numbers in a data setwith first digit is necessarily a rational number, whereas is irrational. log However, we may conceptualize a “Benford” data set of size as being a random sample, i.e. independent realizations of a random variable , such that eq. (2.1) holds with Pr interpreted as “probability.” This is one of the properties of a Benford random variable.The First Digit Law is the most famous part of Benford's Law, but it's not the whole law.Benford's Law may be phrased in terms of the joint distribution of the leftmost digits of the numbers in a data set for all , but it is better to phrase it in terms of the “significand” function. To explain this, consider two famous numbers from physics,written in “scientific notation.” Let meters per second, and Joule seconds. (These are approximations to the speed of light invacuum and Planck's constant, respectively.) The two numbers and are the significands of and , and we write and In general, any may be written in the form with and , and both and in this representation of are unique if we adopt the convention that numbers terminating in an infinite string of nines are to be “rounded up,” e.g. becomes The number in this representation of is the (base 10) “significand of ,” and we write Significands in base 10 are most familiar to us, but other bases ourier Analysis and Benford Random Variables, Page 3are of interest. Let . Any number may be written in the form with and (2.2) Both and in this representation are unique if we adopt a “rounding up” convention whenever is an integer, as above, and the “base significand of ,” written , is defined as this . Hence, with and (2.3) Now let be a positive random variable; that is, Assume that is Pr continuous with a probability density function (pdf). : is base Benford Definition 2.1 (or is -Benford) if and only if the cumulative distribution function (cdf) of is given by Pr log for all (2.4)It's easy to show that a base 10 Benford random variable satisfies the First Digit Lawgiven by eq. (2.1). Let denote the first (most significant) digit in the decimal representation of , and observe that iff for every It's useful at this point to introduce some non-standard notation . Let and recall that the “floor” of , written , is defined as the largest that is less than or equal to integer . Define as: (2.5)and note that for every . If one may interpret as the fractionalpart of , but if this description would be misleading. For example, , so , but , so If we take the logarithm base of eq. (2.3) we obtain log log (2.6)On the other hand, log log log (2.7)As is necessarily an integer and , comparison of eqs. (2.6) log log and (2.7) shows thatlog log log and (2.8)for any Using this notation, we may rephrase Definition 2.1 in several logically equivalent ways: is -Benford iff Introduced into the literature, to the best of my knowledge, by Stein and Shakarchi [9], page 106. ourier Analysis and Benford Random Variables, Page 4(1) for every(2) for every (3)(4) where Pr log log logPr loglog where the notation “ ” means that is uniformly distributed on the half open interval For any random variable , if we sometimes say that is “uniformly distributed modulo one,” abbreviated “u. d. mod 1.” Hence is -Benford iff is log u. d. mod 1.Given a positive random variable and , we may ask (1) whether or not is -Benford; and if it's not, (2) to what extent does it deviate from “Benfordness.” I call aninvestigation of this sort a “Benford analysis.” Let's establish some nomenclature for aBenford analysis. First, define log ln ln where (2.9)Next, let and denote the cdf and pdf of , and denote the cdf and pdf of . Given we may answer the two questions given above. (1) is -Benford iff for almost all (2) If is not -Benford, we may measure its deviation from
Benfordness by any measure of the deviation of from a uniform distribution. For example, if is continuous, or if its only discontinuities are “jumps,” we could use the infinity norm: sup (2.10) We need a way to find from . Under a reasonable assumption, it may be shown that (2.11)for all To see this, and to explain the “reasonable assumption,” begin with the cdf of . For any I frequently use the notation “ Fam parms ” to indicate that the random variable is distributed as a member of a family of random variables, and specified by certain parameters. For example, means that is distributed normally with mean and variance , and Lognormal means that is distributed as a lognormal random variable with parameters and . ourier Analysis and Benford Random Variables, Page 5 Pr Pr (2.12)If has finite support, we may replace the series in eq. (2.12) by a finite sum for some and term by term differentiation yields (2.13)Similarly, simple term by term differentiation of eq. (2.12) yields eq. (2.11). The problemwith this “proof” is that term by term differentiation of an infinite series is not always valid.The condition that's sufficient to justify this procedure is that the series on the right-handside of eq. (2.11) converge . While it's certainly uniformly (Rudin [7], Theorem 7.17)possible to impose weak conditions on that imply the uniform convergence of the series in eq. (2.11), these conditions are somewhat artificial, and (in my opinion) it's better just toassume that this condition holds.Although eq. (2.11) is fundamental for a Benford analysis of , it is not very useful for finding the answers to some analytical questions one may ask. For example, suppose that Lognormal , so As we'll show in log lnSection 4, is not -Benford, but the deviation is generally quite small, so a Lognormal random variable is almost -Benford. One may wish to investigate how this deviation varies as a function of and . For a second example, suppose that is -Benford. We may ask: what is the set is -Benford ? This set certainly includes , but whether it includes any is unclear at this point. Equation (2.11) does not seem very useful for an investigation into these questions.Fourier analysis provides the tools needed to answer questions like these. The Fourierseries representation of is (2.14)where the Fourier coefficients for all are given by (2.15)At first sight, these expressions must look very unpromising for two reasons, one The sense of the equality in eq. (2.14) is explained in Section 3. ourier Analysis and Benford Random Variables, Page 6superficial and the second significant. The superficial reason is that the complexexponentials cos sin are complex-valued, so the series of real-valued functions of eq. (2.11) has been rewrittenas a series of complex-valued functions multiplied by complex coefficients. This may seemlike one step forward and two steps back, but as we'll see in the next section, eq. (2.14) canalways be rewritten as a series of real-valued functions multiplied by real-valuedcoefficients. The significant reason is that computation of the Fourier coefficients given byeq. (2.15) apparently requires that we already know , the function for which we're trying to find a useful expression. As it happens, Fourier transforms come to our rescue here. Wemay show that (2.16)where denotes the Fourier transform of . In words, we may obtain the Fourier coefficients transform of from the Fourier of . This is the key result that allows our Fourier analysis to proceed. All one needs to know to understand the following proof is thedefinition of “the Fourier transform of ”: for any , (2.17)Also, this proof makes implicit use of the uniform convergence of the series in eq. (2.11)(Rudin [7], Theorem 7.16). Proof (of eq. (2.16)). as was to be shown.Combining eqs. (2.11), (2.14), and (2.16), we obtain (2.18)This equation is known as the Poisson summation formula (Stein and Shakarchi [9], page154).We pause, then, in our exposition about Benford analyses, to list the essential facts aboutFourier series and Fourier transforms that are required to continue the argument. Thereader who is familiar with Fourier series and Fourier transforms may skip directly toSection 4.ourier Analysis and Benford Random Variables, Page 7
3. Fourier Series and Fourier Transforms
Elements of are functions from into , but we will mainly be concerned with real valued functions. The space is a Hilbert space, with an inner product defined as (3.1)for any and in , where the overbar denotes complex conjugation. This inner product defines a : for any , norm (3.2)It is known that the collection of complex exponentials where (3.3)forms a complete orthonormal basis for . That is, for any there exists a doubly infinite sequence of (generally complex) scalars such that (3.4)where equality in eq. (3.4) is “in the sense of the norm,” i.e.,lim (3.5)(There may be points where equality in eq. (3.4) doesn't hold. For example, it will not generally hold at points where is not continuous. Equation (3.5), however, implies that the set of points where equality does not hold has measure zero .) The expression on the right-hand side of eq. (3.4) is the “Fourier series expansion” for . The scalars are the “Fourier coefficients” of , and are given by (3.6)Now assume that is real valued. Equations (3.4) and (3.6) state that can be written as a series whose terms are complex valued functions multiplied by complex scalars . Complex exponentials are algebraically convenient, but using them imposes the cost ofworking with complex functions and complex coefficients. However, it may be shown that Technically, eq. (3.2) is a only a “seminorm.” The usual way to get around this difficulty is to redefineequality for elements of . If and are elements of , we say and are and write equal iff has measure zero. This innocuous sounding statement is actually a nontrivial theorem that was first proven by LennartCarleson in 1966. ourier Analysis and Benford Random Variables, Page 8eq. (3.4) can always be rewritten as a series with real valued functions and real coefficients.To be specific, cos sin (3.7)where cossin (3.8)To prove these formulas, first note that , then show that (3.9)for all . Finally, use these formulas to show that cos sin for all . Equations (3.7) and (3.8) embody the “classical Fourier series” expansion for real valued functions in . In practice, it's often useful to rewrite eq. (3.7) in the form cos (3.10)where satisfies (3.11)and is any solution to cos sin and (3.12)The parameters and are not uniquely determined by eqs. (3.11) and (3.12), but in practice natural candidates for and often present themselves. We'll see an example of this shortly.Equation (3.10) says that may be written as a constant plus a series whose components are oscillatory functions of the form cos .For each , the parameter is the “amplitude” of the oscillation and is a “phase.” As varies from to , the function oscillates through exactly cos ourier Analysis and Benford Random Variables, Page 9 cycles. Hence, if represents time (measured, for example, in seconds), then may be interpreted as representing , i.e., cycles per second. frequency If the function is a pdf, , so eq. (3.10) may be written cos (3.13)Hence, is the pdf of a random variable iff for all Furthermore, it follows from eq. (3.9) that (3.14)for all Hence we've proven the following important fact. . Assume Proposition 3.1 that is a pdf on . Then is the pdf of a random variable iff for all We now turn to the topic of Fourier transforms. Fourier transforms of a function aredefined for complex valued functions, but we are mainly interested in this document withfinding the Fourier transforms of real valued functions. In particular, we are interested inthe Fourier transforms of probability density functions. In what follows, I'll give thedefinitions and major properties of Fourier transforms, then describe the simplificationsthat follow from restricting these functions to being real valued.Let and suppose that . Several different definitions of the “Fourier transform” of exist. The definition that is most useful for the purposes of this paper is exp (3.15)where . Hence If is the pdf of a random variable , a convenient way to remember eq. (3.15) is exp (3.16)Perhaps the most common alternative definition of the Fourier transform is exp (3.17)where . For example, this is the way the mathematical software package “Maple” defines the Fourier transform. By the substitution , we see that If is the pdf of a random variable , then there is one more transform of relevance.
The “characteristic function” of is defined as exp exp (3.18)The substitution shows that ourier Analysis and Benford Random Variables, Page 10Both versions of Fourier transform and characteristic functions have some usefulmathematical properties. In what follows I'll list some of these properties for , generally without proofs. My main references for this material are the Wikipedia article onFourier transforms, and Feller's [5] chapter on characteristic functions.The Fourier transform of an integrable function is uniformly continuous. Equation (3.15) implies that (3.19)If is a pdf, eq. (3.19) shows that 1. The Riemann-Lebesgue lemma states that as Let denote the operator that maps into : Inverses . Under weak conditions, all of these transforms have inverses, so may be retrieved from , , and . For the inverse is given by exp . (3.20) Duality . Suppose that and let . Then eq. (3.20) implies that exists and is given by This is called the principle of duality. Linearity . A fundamental property of is that it's a operator. That is, for any linear real numbers and , and for any two functions and in , is also in and (3.21) Shift Property . Define where is a fixed real number. Then (3.22)
Scaling Property . Define where . Then (3.23)
Shift and Scale with Random Variables . Let be a random variable, and let where and are constants with . Then exp expexp exp exp (3.24)Let be the pdf of and be the pdf of . By an argument like that of eq. (3.24) (making ourier Analysis and Benford Random Variables, Page 11use of the linearity of ) one may show that exp (3.25)This is a very useful fact.Before moving on to the next topic, it's convenient to define and rectangular triangular functions. The is defined as unit rectangular function rect if ifif (3.26) The is defined asbasic symmetric triangular function tri (3.27)ifif max
Convolutions . Let and be elements of . The of and , written convolution , is the function defined by the following integral: (3.28)Under these assumptions, is also an element of . For example, suppose that rectThe reader may confirm that triConvolutions are important in probability theory because of the following fact: if is the pdf of and is the pdf of and and are independent random variables, then is the pdf of . For example, if both and are uniformly distributed on , so rect , and and are independent, then the pdf of is tri . Convolution Theorem : if , then (3.29)In words, taking Fourier transforms converts the mathematically complicated operation ofconvolution into the simplier operation of multiplication. (This property holds for othertransforms and for characteristic functions.)Now suppose that is real valued. It follows from eq. (3.15) that .
Equation (3.19) implies that . If for all , then for all .
Fourier transforms are integrals of a complex valued function, but it is easy to rewrite thedefinitions so that the integrals are integrals of real valued functions when is real valued. Specifically,ourier Analysis and Benford Random Variables, Page 12 (3.30)where cos cossin sin . (3.31)Note that is even and is odd. The two integrals that appear in eq. (3.31), sin cos andare known as the “Fourier sine” and “Fourier cosine” transformations, respectively. If is the pdf of a random variable , then and may be rewritten in terms of expected values of real random variables, as follows. cos cossin sin . (3.32)If is even, then is real valued and even. Hence, if is the pdf of , then is real valued and even if is symmetrically distributed around .Appendix B contains a small table of Fourier transforms of probability density functionsadapted from a table in Feller [5], page 503.As a final topic in this section, it is interesting to “compare and contrast” Fouriertransforms and Fourier series. The following table shows the similarities and differences.Transform SeriesDomainCo-domain In both transforms and series, the operator maps a function in one domain into a function in a co-domain. These maps are one-to-one, and the inverse map shows how to retrieve the original function.
4. Benford Analysis Example 1: Lognormal Random Variables
We now resume our study of the Benford analysis of a positive random variable given a base . Recall the following standard notations: first, define . log ln Next, letourier Analysis and Benford Random Variables, Page 13 and denote the cdf and pdf of , and denote the cdf and pdf of .
The Fourier series representation of is (2.14)where the Fourier coefficients for all are given by (2.16)As is a pdf, eq. (2.14) may be rewritten in the form cos (4.1)so cos (4.2)Our first illustration of the use and utility of these constructions is the analysis of lognormalrandom variables.Suppose that Lognormal . As noted in the paragraph following eq. (2.13), it follows that . Hence, to proceed we need to find for random variables of this type. Let denote a standard normal random variable and let denote the pdf of . Let where and are constants with , so . Let denote 's pdf. From the first entry in the table of Fouriertransforms (Appendix B), , and from eq. (3.25) it follows that exp exp exp Putting these facts together, exp exp so exp for all (4.3)where the function is defined as exp (4.4)Hence,ourier Analysis and Benford Random Variables, Page 14 . (4.5)From eq (4.3) we see that is not -Benford. However, as , so is asymptotically -Benford. In fact, becomes quite small for even moderate values of , and it's fair to say in these cases that is “effectively” -Benford. To see this, it's sufficient to note how quickly decreases to zero as increases. When we have , and the upper bound on given in eq. (4.5) is less than or equal to
5. Base Dependence of Benford Random Variables.
My grandfather considered his “law of anomalous numbers” as evidence of a “realworld” phenomenon. He realized that geometric sequences and exponential functions are generally base 10 Benford, and on this basis he wrote (Benford, (1938) [1]):If the view is accepted that phenomena fall into geometric series, then it followsthat the observed logarithmic relationship is not a result of the particularnumerical system, with its base, 10, that we have elected to use. Any other base,such as 8, or 12, or 20, to select some of the numbers that have been suggested atvarious times, would lead to similar relationships; for the logarithmic scales of thenew numerical system would be covered by equally spaced steps by the march ofnatural events. As has been pointed out before, the theory of anomalous numbersis really the theory of phenomena and events, and the but play the poor numbers part of lifeless symbols for living things.This argument seems compelling, and it might seem to apply to Benford randomvariables as well as to geometric sequences and exponential functions. It is thereforesomewhat surprising to observe that a random variable that is base Benford is not generally base Benford when . In this section and the next I examine some of the issues of “base dependence” of Benford random variables.We begin with a definition: following Wójcik [11], the “ ” of a Benford spectrum positive random variable , denoted , is defined as is -Benford . (5.1)The Benford spectrum of may be empty. For example, as shown above, if is a lognormal random variable. In fact, the Benford spectra of essentially all of the usualfamilies of random variables used in statistics (e.g., Gamma, F, Weibull) are empty.There are a couple of elementary propositions about we can prove at this time. See “Benford's Law, A Growth Industry” by Kenneth Ross [6] for further information on this topic ourier Analysis and Benford Random Variables, Page 15
Proposition 5.1 Proof . The set is bounded above. . If is a positive random variable, then Let and denote the pdfs of and ,log ln log ln respectively. Then eq. (3.25) implies that for all .Now suppose that is -Benford. It follows from Proposition 3.1 that 1 As is the Fourier transform of a pdf, it follows that 0 and is continuous.
Hence, there exists such that for all . Therefore ln My proof of the next proposition makes use of the following fact (
Proposition 4.3 (iii),page 44, of Berger and Hill [4] ): a random variable is u. d. mod 1 if and only if is u. d. mod 1 for every integer and . Proposition 5.2 Proof . If then for every . As is -Benford, andHence, for any , andfrom Berger and Hill's Proposition 4.3. As , it follows that . Bibliographic Notes . Wójcik [11] attributes Proposition 5.1 to Schatte (1981) andrediscovered by Whittaker (1983). He attributes Proposition 5.2 to Whittaker.It's appropriate at this point to insert a pair of propositions that are related to Berger andHill's Proposition 4.3.
Proposition 5.3 . If is -Benford, then is -Benford for any constant
Proof . As is -Benford, is u. d. mod 1. But , so log log log log log is u. d. mod 1 from Berger and Hill's Proposition 4.3. Hence is -Benford.We say of this result that Benford random variables are “scale-invariant.” This is only tobe expected if Benford's Law holds for empirical data. For example, one of the variablesincluded in grandfather's data set was “area of rivers.” He doesn't state the in which unit area is measured: it could be acres, or hectares, or something else altogether. Proposition5.3 says that the chosen units don't affect the “Benfordness” of the underlying randomvariable . Proposition 5.4 . Let and denote independent positive random variables. If is -Benford, then the product is -Benford. . Let and denote the
Proof pdfs of , , and , respectively. As and are independent andlog log log ourier Analysis and Benford Random Variables, Page 16log log log , it follows that . From the convolutiontheorem, it follows that for all As is -Benford, for all . Hence for all , and it follows that is -Benford. Proposition 5.4 has an immediate corollary. . If and are
Proposition 5.5 independent positive random variables, then
Suppose that is -Benford. It is of some interest to examine how the distribution of log log varies as a function of . Let denote the pdf of and let denote the pdf of . We know from eq. (4.1) that may be written in the form log cosso we want to examine how each and varies as a function of (and of any parameters that govern the distribution of ). I carried out such a program and reported my results in my 2017 paper [3]. Appendix A of this paper contains a sketch of my methods. Here I willbriefly summarize some of my results. First, it may be shown that the dependence of and on is entirely conveyed through a parameter defined as lnln (5.2)Let be a “seed function” as described in the appendix. Every cdf is a legitimate seed function, and I used the cdfs of standard families of random variables as seed functions togenerate five families of -Benford random variables (which explains the names I gave to these families). For any differentiable seed function , I let . The details of the specification of three of these families are shown in the following table. Family name Parameters Notes Gauss-Benford is the standard normal pdfCauchy-BenfordLapl ace-Benford exp
The function for each of these families is symmetrically distributed around a point , and this fact yields some important analytic simplifications. First, this symmetry impliesthat does not depend on but is given by (5.3)for all . Second, this symmetry implies that the 's all have the form sin (5.4)ourier Analysis and Benford Random Variables, Page 17where varies between the families. The dependence of on and the parameters for these three families is shown in the following table. Family Gauss-BenfordCauchy-BenfordLaplace-Benford expexp
Note that for all of these families. Hence, iff , and this can sin happen for all iff is a positive integer, say . But lnln (5.5)Hence, for all these random variables is precisely the set of integral roots of . Also, note that the parameter affects only the phase, while the parameter affects only . In somewhat more detail, is a decreasing function of .
6. Whittaker's Random Variable .All of the non-empty Benford spectra constructed above are discrete. The question naturally arises: is it possible for a random variable to have a Benford spectrum thatconsists, at least in part, of intervals of positive length? Whittaker [10] showed by anexample that such a random variable exists.Let and let cos (6.1)A plot of between 3 and 3 is shown below. It may be shown that , so is a legitimate pdf. Suppose that and , so log lnln . (6.2)This is . From entry 5 of the table of Fourier transforms in Whittaker's random variable
Appendix B we see that max (6.3)As for all integers , it follows that , and hence that is -Benford.ourier Analysis and Benford Random Variables, Page 18Now let and define log ln ln lnln ln ln (6.4)Let denote the pdf of It follows from eq. (3.25) that maxAs whenever and , it follows that for all integers . Therefore, , and hence is -Benford. Hence, Whittaker's random variable has the remarkable feature that it is -Benford for all . Hence Let's step back for a minute and consider what goes into the construction of a randomvariable with the properties of Whittaker's random variable. The crucial goal is to find apdf whose Fourier transform has bounded support. A reasonable tactic to find is to start with a pdf with bounded support. If the Fourier transform can be rescaled into a legitimate pdf, say , then by duality the Fourier transform is proportional to and therefore has bounded support. What are the requirements on that allow tobe “rescaled into a legitimate pdf”? First, has to be real-valued, which requires that be an even function. Second, has to be non-negative for all . One way to satisfy the second requirement is to require to be the convolution of some even function with itself, as the convolution theorem then guarantees that . The function of Whittaker's random variable was constructed along these lines with equal to rect This is probably the simplest way, but not the only way, to construct a random variable withthe required properties. I think Whittaker's whole construction is remarkably clever.ourier Analysis and Benford Random Variables, Page 19
Appendix A: Seed Functions
The first purpose of this appendix is to describe the use of seed functions to generaterandom variables that are Benford relative to a given base. Let be a -Benford random variable generated by a seed function . A second purpose is to sketch how seed functions may be used to carry out a Benford analysis on where log Definition . A function is a seed function if it satisfies the following three conditions. (1)(2) 1;(3) limlim We say of the last condition that is “unit interval increasing.” If is increasing, then is unit interval increasing, but the converse is not always true. Hence every cdf is a seedfunction, but not every seed function is a cdf.Let be a seed function and define (A.1)The assumption that is unit interval increasing guarantees that for all . Also, it may be shown that (A.2)This is geometrically evident if is an increasing function, as the graph of the function is just the graph of the function shifted to the right by 1. The integral ofeq. (A.2) gives the area between the two graphs as the integral of the distance vertical between the two curves. But the area between the two curves is also equal to the integral ofthe distance between the two curves, which is just horizontal Hence, may be regarded as the pdf of a random variable . Proposition A.1 Proof : If then . For any and , consider (A.3)This sum telescopes, so Letting , we find that ourier Analysis and Benford Random Variables, Page 20 for every As is just the pdf of , this proves the proposition. Corollary . Let . If , then is -Benford. Now let , , , and be defined as above, and let . We are interested in the distribution of and . Let , let denote the pdf of ,log log log and let denote the pdf of . We know that the Fourier series expansion of log is (A.4)From eq. (6.4) we know that , and it follows from eq. (3.25) that (A.5)Hence, we may find the Fourier series expansion of once we've found , defined as Note first that 0 . To evaluate when , assume that is differentiable and let Then integration by parts yields Hence, the Fourier coefficient for any integer is given by (A.6)Equation (A.6) is used to derive eqs. (5.3) and (5.4) as described in my 2017 paper. But wemay observe right away from eq. (A.6) that for all iff either (1) or (2) is an integer. As noted above, is an integer iff for some ourier Analysis and Benford Random Variables, Page 21 Appendix B: A Small Table of Fourier Transforms
Feller [5] gives a table of characteristic functions of selected probability functions. I'veadapted his table to give the Fourier transforms of 8 of his 10 densities. No. Name Density Interval Fourier Transform123 exp sin expcoscos max Notes on this table .(1) This rule has been summarized as “the Fourier transform of a Gaussian is a Gaussian.”(2) If 1, then and sin sin (B.1)when . Now apply eq. (3.25) with and to get rule (2). Letting we obtain rect (B.2) sin(3) Suppose that and , so Let denote the pdf of . Recall that and apply eq. (3.25) to the Fourier sin transform of rule 2 to yield (4) As tri rect rect , it follows from the convolution theorem that tri rect (B.3) sin cosourier Analysis and Benford Random Variables, Page 22(5) The integral of tri over equals , so tri is a pdf. Note that whenever This is important. Feller notes of this result“This formula is of great importance because many Fourier-analytic proofsdepend on the use of a characteristic function vanishing outside a finite interval.”(6) Let be a random variable and denote 's pdf, shown in column 2. Let , define , and let be 's pdf. Then and (B.4)(7) The notation “Bi. Exp.” is an abbreviation for “Bilateral Exponential.” This pdf is alsoknown as that of a Laplace random variable. (8) The Cauchy and Bilateral Exponential distributions are dual to one another.Let be a random variable with pdf given by rule (1), (7), or (8) in this table, let where , and let be 's pdf. Then is distributed symmetricallyaround with a “spread” measured by , and eq. (3.25) with and may be used to find . For rule (1) is the standard deviation of . This is true for the other two distributions. not ourier Analysis and Benford Random Variables, Page 23
References [1] Benford, Frank (1938), The Law of Anomalous Numbers, .,
Proc. Amer. Philosophical Soc
An Introduction to Benford's Law , PrincetonUniversity Press.[5] Feller, William (1971), ,
An Introduction to Probability Theory and Its Applications (Volume II, 2nd ed.), John Wiley & Sons.[6] Ross, Kenneth (2011), “Benford's Law, A Growth Industry,”
Amer. Math. Monthly
Principles of Mathematical Analysis [8] Schatte, Peter (1981), “On Random Variables with Logarithmic Mantissa DistributionRelative to Several Bases,”
Elektronische Informationsverarbeitung und Kybernetik
Fourier Analysis: An Introduction
University Press.[10] Whittaker, James (1983), “On Scale-Invariant Distributions,” .,