The Kolmogorov continuity theorem is a fundamental result in the theory of stochastic processes, particularly in the study of Brownian motion and other continuous-time processes. It provides conditions under which a collection of random variables (typically indexed by time) possesses a continuous version, which means that the sample paths of the process can be modified to be continuous with probability one.

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If a stochastic process indexed by a subset of satisfies
for some , then it has a continuous modification. More precisely, that modification is locally -Hölder continuous for every .