Lévy's continuity theorem

Lévy's continuity theorem states that pointwise convergence of characteristic functions to a function continuous at zero implies weak convergence to a probability distribution having that characteristic function.

Characteristic function: The Fourier transform of a probability distribution, defined as the expected value of e^{itX}. Pointwise convergence is tested on these transforms rather than directly on the distributions.
Probability measure: A measure assigning total mass one to the events of a probability space. The theorem identifies its limiting characteristic function with that of a probability measure.
Central limit theorem: A family of results showing that normalized sums of many independent variables approach a normal distribution under suitable conditions. Characteristic functions provide a common proof strategy, with Lévy’s theorem converting their limit into distributional convergence.
Continuity theorem for characteristic functions: A theorem characterizing weak convergence of probability measures through pointwise convergence of their characteristic functions. This name is often used for the converse direction as well as the implication stated by Lévy’s theorem.
Weak convergence: Convergence of probability measures against every bounded continuous test function. This is the mode of convergence the theorem concludes.
Cumulative distribution function: The function giving the probability that a real random variable is at most each real number. Weak convergence can be checked through these functions at continuity points.
Weak law of large numbers: A result stating that sample averages converge in probability to their expected value under suitable assumptions. Characteristic-function arguments can establish this limit by applying the theorem to the averages.
Prokhorov's theorem: A theorem characterizing relative compactness of probability measures through tightness on suitable spaces. It approaches weak convergence through compactness rather than directly through characteristic functions.
Tightness: A family of probability measures is tight if, for every positive error tolerance, one compact set contains all but that much mass. Tightness prevents probability mass from escaping while characteristic functions converge.
Pointwise convergence: A sequence of functions converges pointwise when it converges at each fixed input. The theorem assumes this convergence for the characteristic functions.

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