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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

Source:
An Introduction to Benford's Law
Author(s):

Arno Berger

Theodore P. Hill

Publisher:
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

List of Symbols

List of Symbols

logarithm base b of

List of Symbols

  • 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

List of Symbols

(x) fractional part of

List of Symbols

  • 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

List of Symbols

  • 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.

List of Symbols

  • 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:

List of Symbols

  • Φ‎ standard normal distribution function:

List of Symbols

  • 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), · · ·)

0

List of Symbols

ο‎.

List of Symbols

List of Symbols

real, imaginary part of

List of Symbols

List of Symbols

conjugate, absolute value (modulus) of

List of Symbols

arg ζ‎ argument of

List of Symbolsarg

List of Symbolsand

List of SymbolsS unit circle in

List of Symbols

  • spanQ^ rational span of Ζ‎ C C;

List of Symbols

  • 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

List of Symbols

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

List of Symbols

p(A) spectral radius of

List of Symbols

List of Symbols

Euclidean norm of

List of Symbols

List of Symbols

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

List of Symbols

(Xn) converges in distribution to X.

List of Symbols

(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

List of Symbols

end of PROOF

List of Symbols

end of EXAMPLE