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Power Laws for Monkeys Typing Randomly: The Case of Unequal Probabilities

An early result in the history of power laws, due to Miller, concerned
the following experiment. A monkey types randomly on a keyboard
with $N$ letters $(N >> 1)$ and a space bar, where a space separates
words. A space is hit with probability $p$; all other letters are hit
with equal probability $(1-p)/N$. Miller proved that in this
experiment, the rank-frequency distribution of words follows a power
law. The case where letters are hit with unequal probability has been
the subject of recent confusion, with some suggesting that in this
case the rank-frequency distribution follows a lognormal
distribution. We prove that the rank-frequency distribution follows a
power law for assignments of probabilities that have rational
log-ratios for any pair of keys, and we present an argument of
Montgomery that settles the remaining cases, also yielding a power
law. The key to both arguments is the use of complex analysis. The
method of proof produces simple explicit formulas for the coefficient
in the power law in cases with rational log-ratios for the assigned
probabilities of keys. Our formula in these cases suggests an exact
asymptotic formula in the cases with an irrational log-ratio, and this
formula is exactly what was proved by Montgomery.