Kolmogorov continuity theorem

The Kolmogorov continuity theorem gives conditions on a stochastic process’s increment moments that guarantee a continuous modification, often with a quantitative Hölder regularity bound.

Hölder continuity: A function is Hölder continuous if its increments are bounded by a constant times a positive power of distance. The theorem’s conclusion is typically a modification whose paths satisfy a Hölder bound.
Stochastic process: A family of random variables indexed by time, space, or another parameter. The theorem applies to this family and its sample paths.
Brownian motion: A continuous-time process with independent Gaussian increments, starting at zero and having variance proportional to elapsed time. Its increment moments satisfy the criterion, which establishes a continuous version.
Kolmogorov tightness criterion: A moment condition on process increments that can establish tightness of probability laws on a function space. It resembles the continuity theorem but controls families of laws rather than one process’s modification.
Markov inequality: A probability bound that limits the chance a nonnegative random variable exceeds a threshold using its expectation. Moment estimates become tail bounds through Markov’s inequality.
Moment (probability theory): An expectation of a power of a random variable, such as its variance or fourth moment. The hypothesis bounds moments of differences between process values.
Gaussian process: A collection of random variables whose every finite subcollection has a joint Gaussian distribution. Increment bounds provide a standard route to continuous versions of Gaussian processes.
Arzelà–Ascoli theorem: A compactness theorem characterizing relatively compact families of continuous functions through boundedness and equicontinuity. It concerns compactness of function families, while Kolmogorov’s criterion derives path regularity probabilistically.
Borel–Cantelli lemma: A probability theorem stating conditions under which events occur only finitely often almost surely. Applying it across finer grids turns increment tail bounds into almost-sure path control.
Metric space: A set equipped with a distance function satisfying nonnegativity, symmetry, and the triangle inequality. Distances between parameter values determine the scale in the increment bound.

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