Portmanteau theorem

The Portmanteau theorem gives equivalent conditions for convergence in distribution of probability measures, including inequalities for probabilities of closed and open sets.

Weak convergence of measures: A mode of convergence in which integrals against every bounded continuous function converge. The theorem characterizes this convergence through equivalent conditions.
Probability measure: A measure assigning total mass one to a measurable space. The theorem compares probability measures and their mass on measurable sets.
Continuous mapping theorem: A theorem stating that continuous transformations preserve convergence in distribution. It often combines with Portmanteau tests to establish convergence of transformed random variables.
Almost sure convergence: Convergence of random variables that occurs with probability one. It is stronger than convergence in distribution and is not what the theorem characterizes.
Convergence in distribution: Convergence of probability laws, defined by convergence of distribution functions at continuity points of the limit. This is the probabilistic convergence notion the theorem characterizes.
Borel measure: A measure defined on the Borel sets of a topological space. The theorem’s topological conditions are expressed through Borel sets.
Slutsky's theorem: A theorem describing limits of sums, products, and ratios involving convergent random variables. Portmanteau characterizations underpin distributional limit arguments used in its proof.
Convergence in probability: Convergence in which the probability of deviating from the limit by a fixed amount tends to zero. It is stronger than distributional convergence when the limit is constant.
Bounded continuous function: A continuous function whose absolute value is bounded across its domain. Convergence of integrals against these functions is one of the theorem’s defining conditions.
Topology: The mathematical study of open sets, continuity, and limits in spaces. Open and closed sets supply the theorem’s geometric convergence tests.

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