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History-dependent processes are ubiquitous in natural and social systems.Many such stochastic processes, especially those that are associated withcomplex systems, become more constrained as they unfold, meaning that theirsample-space, or their set of possible outcomes, reduces as they age. Wedemonstrate that these sample-space reducing (SSR) processes necessarily leadto Zipf's law in the rank distributions of their outcomes. We show that byadding noise to SSR processes the corresponding rank distributions remain exactpower-laws, $p(x)\sim x^{-\lambda}$, where the exponent directly corresponds tothe mixing ratio of the SSR process and noise. This allows us to give a precisemeaning to the scaling exponent in terms of the degree to how much a givenprocess reduces its sample-space as it unfolds. Noisy SSR processes furtherallow us to explain a wide range of scaling exponents in frequencydistributions ranging from $\alpha = 2$ to $\infty$. We discuss severalapplications showing how SSR processes can be used to understand Zipf's law inword frequencies, and how they are related to diffusion processes in directednetworks, or ageing processes such as in fragmentation processes. SSR processesprovide a new alternative to understand the origin of scaling in complexsystems without the recourse to multiplicative, preferential, or self-organisedcritical processes.

Understanding scaling through history-dependent processes with collapsing sample space