## 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

Source:
An Introduction to Benford's Law
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

with 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

logarithm base b of

• In χ‎ natural logarithm (base e) of

• 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

(x) fractional part of

• 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

• 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

• 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.

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

• Β‎ Benford distribution on (R+, §)

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

• Φ‎ standard normal distribution function:

• 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

ο‎.

real, imaginary part of

conjugate, absolute value (modulus) of

arg ζ‎ argument of

arg

and

S unit circle in

• spanQ^ rational span of Ζ‎ C C;

• 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

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

p(A) spectral radius of

Euclidean norm of

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

(Xn) converges in distribution to X.

(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:

• P* finitely additive probability measure on Ω‎ C M

end of PROOF

end of EXAMPLE