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An Introduction to Benford's Law$

Arno Berger and Theodore P. Hill

Print publication date: 2015

Print ISBN-13: 9780691163062

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691163062.001.0001

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(p.231) List of Symbols

(p.231) List of Symbols

An Introduction to Benford's Law

Arno Berger

Theodore P. Hill

Princeton University Press

  • N, No, Ζ‎ set of positive integers, nonnegative integers, integers

  • Q,R+,R, C set of rational, positive real, real, complex numbers

  • [a, b) (half-open) interval

    List of Symbolswith a < b.

  • D\,D-2, etc. first, second, etc. significant decimal digit

mt h significant digit base 6, b > 2 log χ‎ logarithm base 10 of

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logarithm base b of

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  • In χ‎ natural logarithm (base e) of

    List of Symbols

  • Δ‎ deviation (in percent) from first-digit law:

Δ‎ = 100 ·

  • Φ‎ A cardinality (number of elements) of finite set A.

  • U(a, b) random variable uniformly distributed on (a, b) with a < b.

  • S (decimal) significand function

[x\ largest integer not larger than

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(x) fractional part of

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  • A σ‎-algebra, on some non-empty set Ω‎

  • σ‎(£) σ‎-algebra generated by collection Ε‎ of subsets of Ω‎

  • 23 Borel σ‎-algebra on R or parts thereof

    σ‎(/) σ‎-algebra generated by function / : Ω‎ —> R § significand σ‎-algebra; § = R+ Π‎ σ‎ (S).

    (Ω‎, Α‎, Ρ‎) probability space

Ac complement of A in set

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  • A\B set of elements in A but not in B; A\B = An Bc.

  • (p.232) ΑΔΒ‎ symmetric difference of A and Β‎; ΑΔΒ‎ = (A\B) U (B\A) λ‎ Lebesgue measure on (R, Έ‎) or parts thereof

  • λα‎6 normalized Lebesgue measure (uniform distribution) on ([α‎,5),Έ‎[α‎,5))

  • δω‎ Dirac probability measure concentrated at

    List of Symbols

  • p(C) (natural) density ofCcN

  • 1A indicator function of set A.

  • (Fn) sequence of Fibonacci numbers; (Fn) = (1,1, 2, 3, 5,…)

  • (pn) sequence of prime numbers; (pn) = (2, 3, 5, 7,11,…)

  • i.i.d. independent, identically distributed

  • X, Y,… (real-valued) random variables Ω‎ —> R

  • K[X] expected (or mean) value of random variable X.

  • varX variance of random variable X with E[|X|] < +∞: varX = E[(X–EX)2 ]

  • Ρ‎ probability measure on (R, IB), possibly random

  • Ρχ‎ distribution of random variable X.

  • Fp, Fx distribution function of Ρ‎, Χ‎.

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  • kth Fourier coefficient of probability Ρ‎ on S[0,1):

List of Symbols

  • Β‎ Benford distribution on (R+, §)

  • Δοο‎ deviation (100 χ‎ sup-norm) from Benford's law:

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  • Φ‎ standard normal distribution function:

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  • Vf, P o / " 1 probability measure on R induced by Ρ‎ and measurable

  • function / : Ω‎ –• R, via P; ( · ) = F(7^1 (·))

  • u.d. mod 1 uniformly distributed modulo one

  • a.a. (Lebesgue) almost all

  • a.s. almost surely, i.e., with probability one

  • y, \ increasing, decreasing (not necessarily strictly)

  • (p.233) (/"(xo)) orbit of xo (under map /); (/»(*<,)) = (/(zo), /o/(*o), · · ·)


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List of Symbols

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real, imaginary part of

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conjugate, absolute value (modulus) of

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arg ζ‎ argument of

List of Symbolsarg

List of Symbolsand

List of SymbolsS unit circle in

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  • spanQ^ rational span of Ζ‎ C C;

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  • Ck set of all k times continuously differentiable functions

  • C°° set of all smooth (i.e., infinitely differentiable) functions

  • Ng Newton map associated with differentiable function g.

  • ~Rdxd set (linear space) of all real d χ‎ d-matrices

  • Id d χ‎ d-identity matrix

  • La set (linear space) of all linear observables on Rd x d

[A]jk entry in j t h row, kth column of

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σ‎ (A) spectrum (set of all eigenvalues) of

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p(A) spectral radius of

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Euclidean norm of

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first, second, kth derivative of χ‎ = x(t) with respect to t.

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(Xn) converges in distribution to X.

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(Xn) converges to X almost surel.

  • r.p.m. random probability measure

  • EP expectation of r.p.m. Ρ‎.

  • T + set of all positive terminating decimals:

    List of Symbols

  • P* finitely additive probability measure on Ω‎ C M

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end of PROOF

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end of EXAMPLE