We characterize the conditions under which rank-based systems of continuous semimartingales generate an asymptotic size distribution that satisfies Zipf's law. For a system that follows the strong form of Gibrat's law, with growth rates and volatilities that do not vary across rank, these conditions require that the system be conservative and complete, and are satisfied by many large systems of time-dependent ranked observations. We generalize Zipf's law to a less restrictive form in which a log-log plot of size versus rank does not have to be a straight line of slope -1, but rather has a tangent line of slope -1 at some point. Under certain regularity conditions, we show that the same conditions of conservation and completeness imply that rank-based systems that deviate from Gibrat's law in a specific but realistic manner generate an asymptotic size distribution that is quasi-Zipfian. Because many real-world systems that follow the strong form of Gibrat's law satisfy Zipf's law, and even more systems that do not follow the strong form of Gibrat's law are quasi-Zipfian, our results offer an explanation for the universality of Zipf's law.
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