Zeta distribution

zeta
	Probability mass function; Plot of the Zeta PMF on a log-log scale. (The function is only defined at integer values of k. The connecting lines do not indicate continuity.)
	Cumulative distribution function
Parameters
Support
pmf
CDF
Mean
Mode
Variance
Entropy
MGF
CF

In probability theory and statistics, the zeta distribution is a discrete probability distribution. If X is a zeta-distributed random variable with parameter s, then the probability that X takes the integer value k is given by the probability mass function

f_{s}(k)=k^{{-s}}/\zeta (s)\,

where ζ(s) is the Riemann zeta function (which is undefined for s = 1).

The multiplicities of distinct prime factors of X are independent random variables.

The Riemann zeta function being the sum of all terms $k^{{-s}}$ for positive integer k, it appears thus as the normalization of the Zipf distribution. The terms "Zipf distribution" and the "zeta distribution" are often used interchangeably. But note that while the Zeta distribution is a probability distribution by itself, it is not associated to the Zipf's law with same exponent. See also Yule–Simon distribution

Definition

The Zeta distribution is defined for positive integers $k\geq 1$ , and its probability mass function is given by

P(x=k)={\frac {1}{\zeta (s)}}k^{-s}

,

where $s>1$ is the parameter, and $\zeta (s)$ is the Riemann zeta function.

The cumulative distribution function is given by

P(x\leq k)={\frac {H_{k,s}}{\zeta (s)}},

where $H_{k,s}$ is the generalized harmonic number

H_{k,s}=\sum _{i=1}^{k}{\frac {1}{i^{s}}}.

Moments

The nth raw moment is defined as the expected value of Xⁿ:

m_{n}=E(X^{n})={\frac {1}{\zeta (s)}}\sum _{{k=1}}^{\infty }{\frac {1}{k^{{s-n}}}}

The series on the right is just a series representation of the Riemann zeta function, but it only converges for values of s-n that are greater than unity. Thus:

m_{n}=\left\{{\begin{matrix}\zeta (s-n)/\zeta (s)&{\textrm {for}}~n<s-1\\\infty &{\textrm {for}}~n\geq s-1\end{matrix}}\right.

Note that the ratio of the zeta functions is well defined, even for n > s − 1 because the series representation of the zeta function can be analytically continued. This does not change the fact that the moments are specified by the series itself, and are therefore undefined for large n.

Moment generating function

The moment generating function is defined as

M(t;s)=E(e^{{tX}})={\frac {1}{\zeta (s)}}\sum _{{k=1}}^{\infty }{\frac {e^{{tk}}}{k^{s}}}.

The series is just the definition of the polylogarithm, valid for $e^{t}<1$ so that

M(t;s)={\frac {\operatorname {Li}_{s}(e^{t})}{\zeta (s)}}{\text{ for }}t<0.

The Taylor series expansion of this function will not necessarily yield the moments of the distribution. The Taylor series using the moments as they usually occur in the moment generating function yields

\sum _{{n=0}}^{\infty }{\frac {m_{n}t^{n}}{n!}},

which obviously is not well defined for any finite value of s since the moments become infinite for large n. If we use the analytically continued terms instead of the moments themselves, we obtain from a series representation of the polylogarithm

{\frac {1}{\zeta (s)}}\sum _{{n=0,n\neq s-1}}^{\infty }{\frac {\zeta (s-n)}{n!}}\,t^{n}={\frac {\operatorname {Li}_{s}(e^{t})-\Phi (s,t)}{\zeta (s)}}

for $\scriptstyle |t|\,<\,2\pi$ . $\scriptstyle \Phi (s,t)$ is given by

\Phi (s,t)=\Gamma (1-s)(-t)^{{s-1}}{\text{ for }}s\neq 1,2,3\ldots

\Phi (s,t)={\frac {t^{{s-1}}}{(s-1)!}}\left[H_{s}-\ln(-t)\right]{\text{ for }}s=2,3,4\ldots

\Phi (s,t)=-\ln(-t){\text{ for }}s=1,\,

where H_s is a harmonic number.

The case s = 1

ζ(1) is infinite as the harmonic series, and so the case when s = 1 is not meaningful. However, if A is any set of positive integers that has a density, i.e. if

\lim _{n\to \infty }{\frac {N(A,n)}{n}}

exists where N(A, n) is the number of members of A less than or equal to n, then

\lim _{s\to 1^{+}}P(X\in A)\,

is equal to that density.

The latter limit can also exist in some cases in which A does not have a density. For example, if A is the set of all positive integers whose first digit is d, then A has no density, but nonetheless the second limit given above exists and is proportional to

\log(d+1)-\log(d)=\log \left(1+{\frac {1}{d}}\right),\,

which is Benford's law.

Infinite divisibility

The Zeta distribution can be constructed with a sequence of independent random variables with a Geometric distribution. Let $p$ be a prime number and $X(p^{-s})$ be a random variable with a Geometric distribution of parameter $p^{{-s}}$ , namely

$\quad \quad \quad \mathbb {P} \left(X(p^{-s})=k\right)=p^{-ks}(1-p^{-s})$

If the random variables $(X(p^{-s}))_{p\in {\mathcal {P}}}$ are independent, then, the random variable $Z_{s}$ defined by

$\quad \quad \quad Z_{s}=\prod _{p\in {\mathcal {P}}}p^{X(p^{-s})}$

has the Zeta distribution : $\mathbb {P} \left(Z_{s}=n\right)={\frac {1}{n^{s}\zeta (s)}}$ .

Stated differently, the random variable $\log(Z_{s})=\sum _{p\in {\mathcal {P}}}X(p^{-s})\,\log(p)$ is infinitely divisible with Lévy measure given by the following sum of Dirac masses :

$\quad \quad \quad \Pi _{s}(dx)=\sum _{p\in {\mathcal {P}}}\sum _{k\geqslant 1}{\frac {p^{-ks}}{k}}\delta _{k\log(p)}(dx)$

External links

Gut, Allan. "Some remarks on the Riemann zeta distribution". CiteSeerX 10.1.1.66.3284. What Gut calls the "Riemann zeta distribution" is actually the probability distribution of −log X, where X is a random variable with what this article calls the zeta distribution.
Weisstein, Eric W. "Zeta distribution". MathWorld.

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Probability distributions
List
Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot
Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta
Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	Degenerate Dirac delta function Singular Cantor
Families	Circular compound Poisson elliptical exponential natural exponential location–scale maximum entropy mixture Pearson Tweedie wrapped

Probability mass function Plot of the Zeta PMF on a log-log scale. (The function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Cumulative distribution function
Parameters	$s\in (1,\infty )$
Support	$k\in \{1,2,\ldots \}$
pmf	${\frac {1/k^{s}}{\zeta (s)}}$
CDF	${\frac {H_{{k,s}}}{\zeta (s)}}$
Mean	${\frac {\zeta (s-1)}{\zeta (s)}}~{\textrm {for}}~s>2$
Mode	$1\,$
Variance	${\frac {\zeta (s)\zeta (s-2)-\zeta (s-1)^{2}}{\zeta (s)^{2}}}~{\textrm {for}}~s>3$
Entropy	$\sum _{{k=1}}^{\infty }{\frac {1/k^{s}}{\zeta (s)}}\log(k^{s}\zeta (s)).\,\!$
MGF	${\frac {\operatorname {Li}_{s}(e^{t})}{\zeta (s)}}$
CF	${\frac {\operatorname {Li}_{s}(e^{{it}})}{\zeta (s)}}$