What is the discrete equivalent of the powerlaw distribution?

The question is all in the title. The only background I can add, is that I am finding a powerlaw to interpolate beautifully between my data points. Only that my data appears in integers only, so I would better use the discrete equivalent of the powerlaw distribution - that is to say, a probability density function equal to $ax^{-b}$ with $a$ and $b$ positive, and $x > 1$ - to use it for a Monte Carlo simulation. From a quick googling it seems like a Zeta distribution might be what I need, but I am unsure.

• Can you make it more precise what do you mean by powerlaw distribution? – Tim Jan 4 '17 at 10:45
• Thanks Tim - just edited the question. Do not know yet how to use Latex in CrossValidated though – famargar Jan 4 '17 at 10:49
• You haven't defined the domain of $x$ though, nor what $ax^{-b}$ represents (the pmf? the survivor function? something else?) so your definition is incomplete (and different authors use different definitions!). On using the subset of LaTeX implemented in MathJax, see here. – Glen_b Jan 4 '17 at 11:27
• Thanks Glen, hope my last edit contains all relevant infos. – famargar Jan 4 '17 at 11:54

If you want a discrete distribution where $P(X=x)\propto x^{-\alpha}$ for $x=1,2,3,...$ then you would have a zeta distribution.

If you want a discrete distribution where $P(X=x)\propto x^{-\alpha}$ for $x=1,2,3,...,N$ then you would have a truncated zeta distribution, also known as Zipf's law.

On the other hand, if (as at least some authors do) you define a power law up to a slowly varying function -- i.e. where say

$P(X=x) = L(x)\cdot x^{-\alpha}\,,\quad x=1,2,3,...$

or

$P(X\gt x) = L(x)\cdot x^{-\alpha}\,,\quad x=1,2,3,...$

then neither of these is a zeta unless $L$ is the identity. Indeed, this definition encompasses an infinity of distributions