[Zipf's law](http://en.wikipedia.org/wiki/Zipf%27s_law) is the empirical observation that in many real-life populations of n objects, the $k^{th}$ largest object has size proportional to $1/k$, at least for $k$ significantly smaller than $n$ (and one also sometimes needs to assume $k$ somewhat larger than 1). It is a special case of a power law distribution (in which $1/k$ is replaced with $1/k^\alpha$ for some exponent $\alpha$), but the remarkable thing is that in many empirical cases (e.g. frequencies of words, or sizes of cities), the exponent is very close to 1.


My question is: is there a "natural" random process (e.g. a birth-death process) that one can rigorously demonstrate (or at least conjecture) to generate populations of n non-negative quantities $X\_1,\ldots,X\_n$ (with n large but possibly variable) that obey Zipf's law on average with exponent 1? There are plenty of natural ways to generate processes that have power law tails (e.g. consider n positive quantities $X\_1,\ldots,X\_n$ evolving by iid copies of log-Brownian motion), but I don't see how to ensure the exponent is 1 without artificially setting the parameters to force this. 


Ideally, such processes should be at least somewhat plausible as models for an empirical situation in which Zipf's law is observed to hold, such as city sizes, but I would be happy with any non-artificial example of a process.


One obstruction here is the exponent one property is not invariant with respect to taking powers: if $X\_1,\ldots,X\_n$ obeys Zipf's law with exponent one, then for any fixed $\beta>0$, $X\_1^\beta,\ldots,X\_n^\beta$ obeys the power law with a different exponent $\beta$. So whatever random process one would propose for Zipf's law must somehow be quite different from its powers.



Is there a natural random process that is rigorously known to produce Zipf's law?
I'm not sure if this is an "answer" to your question, but I recall seeing somewhere that someone had shown that if you create a document by selecting the characters a...z plus a space character with uniform frequency then the "words" of such a document have a frequency distribution that follows Zipf's Law. (A little anecdote: when I was an undergraduate, I took a course on "Inductive Logic" given by Zipf. I recall being rather annoyed because he spent a lot of the time lecturing about "his" law and having us form groups that as part of our class work collected statistics to test it :-) 


(Added Remarks) I recalled that when we tested Zipf's Laws for city populations back then (more than 50 years ago !) the results were quite good---i.e., the population of the n-th city was pretty close to $1/n$ times the population of the first for many countries. I decided to see if that was still so. For the US it pretty much is:


<http://www.infoplease.com/ipa/A0763098.html#axzz0zuwyduxq>


However, for China, it is WAY off---not even close:


<http://en.wikipedia.org/wiki/List_of_cities_in_the_People%27s_Republic_of_China_by_population>


Of course the population of Chinese cities has been changing rapidly due to migrations into them from the countryside, and perhaps Zipf's Law pertains only to stable situations when things are in equilibrium.


