Comparing three discrete power laws

Yesterday I wrote about the zeta distribution and the Yule-Simon distribution and said that they, along with the Zipf distribution, are all discrete power laws. This post fills in detail for that statement.

The probability mass functions for the zeta, Zipf, and Yule-Simon distributions are as follows.

\begin{align*} f_\zeta(k; s) &= k^{-s} / \zeta(s) \\ f_y(k; \rho) &= \rho B(k, \rho+1) \\ f_z(k; N, s) &= k^{-s} / H_{N,s} \end{align*}

Here the subscripts ζ, y, and z stand for zeta, Yule, and Zipf respectively. The distribution parameters follow after the semicolon.

Comparing Zipf and zeta

The Zipf distribution is only defined on the first N positive integers. The normalizing constant for the Zipf distribution is the Nth generalized harmonic number with exponent s.

H_{N,s} = \sum_{k=1}^N k^{-s}

As N goes to infinity, HN,s converges to ζ(s); this is the definition of the ζ function. So the Zipf and zeta distributions are asymptotically equal.

Comparing zeta and Yule

This post showed that for large k,

f_y(k; \rho) \approx \rho \Gamma(\rho + 1) \frac{1}{(k+\rho)^{\rho + 1}}

That is, fy(k, ρ) is proportional to a power of k, except k is shifted by an amount ρ.

To compare the zeta and Yule distributions we’ll need to compare zeta with s+1 to Yule with ρ in order to make the exponents agree, and we’ll need to shift the zeta distribution by s+1.

When we plot the ratio of the pmfs, we see that the distributions agree in the limit as k gets large.

In this plot s = ρ = 1.5.

The ratio is approaching a constant, and in fact the limit is

\frac{1}{ \rho \,\Gamma(\rho + 1)\,\zeta(s+1)}

based on the ratio of the proportionality constants.


Note that the three distributions are asymptotically proportional, given the necessary shift in k, but in different ways. The Zipf distribution converges to the zeta distribution, for every k, as N goes to infinity. The zeta and Yule distributions are asymptotically proportional as k goes to infinity. So one proportion is asymptotic in a parameter N and one is asymptotic in an argument k.

One thought on “Comparing three discrete power laws

  1. It would be helpful if you added some examples of usage to clarify how these distributions arise. As an example: extreme value distributions are sometimes used to model breaking strengths of engineered materials.

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