Q:

Do more closely related languages, have more similar Zipfian distributions?

"Zipf's law" is just a pretentious way of saying that many types of data, in various sciences, fit certain kinds of power law distribution. E.g. in linguistics, for a corpus of English word frequency is inversely proportional to the word's rank in frequency (the most common word occurs twice as often as the second most common word, three times as often as the third most common word, etc.). I'm assuming that letter frequency, bigram frequency, trigram frequency, phoneme frequency, etc. are also power-law distributed.
What I'm asking isn't especially to do with the distributions being power laws. I want to know: are the rankings of word frequencies, and other frequencies, more similar (in whatever precise sense of similarity that linguists and statisticians use), between languages which are more closely related?(E.g. are more closely related languages, more likely to have the same most-common phoneme?)

A:

There is definitely a correlation, but it is not absolute.  If we imagine a spectrum of values of s (the value of the exponent characterizing the distribution), different language families overlap on the spectrum.
The distribution is a function of morphological typology and orthography, and sometimes otherwise very closely related languages differ fundamentally with regard to these.

Morphological Typology

For example English lost most morphology and gender compared to earlier Germanic languages and generally IE languages.  A famous example for Zipf's law is English the vs Spanish el, la, los, los, del....  As that cycle completes some grammaticalised words may become affixes like English 'll, or syntax may change as was the case with French shifting to non-pro-drop.

Orthography

Hindustani has two scripts.  Chinese has many homographs when written in Chinese, but many homophones when written in Pinyin.  In some Hebrew corpora, vowels are indicated, in others only when necessary, resulting in more homographs.  German writes noun compounds together, but modern English usually does not.  The space character was an innovation that is still not universal, and the definition of a word or token is a matter of debate.
So English and Chinese may have distributions more similar to each other than English and Celtic, or Chinese and Old Chinese, even though English and Chinese are completely unrelated, whereas the other pairs are in the same family.
That said, some of these typological changes happen because of creolisation or substrate, so there are open questions about whether these languages are not in a different family than their nominal ancestors.  But even if we assume English is a fusion language, it is a fusion of IE languages.
This is all assuming that the corpora used are actually comparable.  English and Scots are very similar languages, but if we use each language's  Wikipedia as a corpus to generate the distribution, then there are other variables.

