Prokhorov's theorem

Prokhorov's theorem characterizes relative compactness of probability measures under weak convergence: on a complete separable metric space, a family is relatively compact exactly when it is tight.

Tightness of measures: A family of measures is tight if, for every positive error tolerance, one compact set contains all but that much mass for every measure. Tightness is the condition Prokhorov’s theorem equates with relative compactness.
Portmanteau theorem: The Portmanteau theorem gives equivalent characterizations of weak convergence of probability measures. Its closed-set inequalities help connect tightness to weak limits.
Andrey Kolmogorov: Andrey Kolmogorov was a Russian mathematician who established foundational axioms and results in modern probability theory. His measure-theoretic framework helped make probability measures central mathematical objects.
Weak convergence of stochastic processes: Weak convergence of stochastic processes is convergence in distribution of their laws on a function space. Tightness criteria help establish subsequential limits for process laws.
Subsequence principle: The subsequence principle studies convergence by showing that every sequence has a subsequence with a desired limiting behavior. Prokhorov’s theorem converts tightness into the subsequences needed for limit identification.
Weak convergence of probability measures: Probability measures converge weakly when their integrals against every bounded continuous function converge. This is the topology in which the theorem’s relative compactness is assessed.
Prokhorov metric: The Prokhorov metric metrizes weak convergence of probability measures on a separable metric space. It expresses weak compactness through metric compactness in standard settings.
Yurii Prokhorov: Yurii Prokhorov was a Soviet mathematician known for foundational work in probability theory and measure convergence. His work established the compactness criterion now bearing his name.
Functional central limit theorem: A functional central limit theorem asserts convergence of random functions or processes to a limiting process, often Brownian motion. Proofs commonly establish tightness before identifying the process limit.
Empirical process theory: Empirical process theory studies random fluctuations of empirical measures and indexed classes of functions. Its weak convergence proofs frequently rely on tightness and compactness of laws.

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