Convergence in distribution

Convergence in distribution is a mode of convergence in which cumulative distribution functions converge at every continuity point of the limiting distribution. It captures convergence of probability laws without requiring random variables to be close on the same probability space.

Cumulative distribution function: A function giving the probability that a random variable is at most each real number. The defining test compares these functions at every continuity point of the limit.
Convergence in probability: A mode of convergence requiring the probability of a fixed-size deviation to vanish. Convergence in probability implies convergence in distribution, but the converse generally fails.
Central limit theorem: A theorem stating that suitably normalized sums of independent variables converge in distribution to a normal law. Its conclusion is a canonical distributional limit for sums of random variables.
Convergence of moments: Convergence of the expected powers or other specified moments of random variables. Convergence of moments does not always determine convergence of distributions without extra conditions.
Weak convergence of probability measures: Convergence of probability measures against every bounded continuous test function. This formulation is equivalent to convergence in distribution on standard spaces such as the real line.
Almost sure convergence: A mode of convergence in which random variables converge outside an event of probability zero. Almost sure convergence is stronger and implies convergence in distribution.
Slutsky's theorem: A theorem describing limits of sums, products, and ratios of convergent random variables. It combines distributional convergence with convergence in probability in asymptotic calculations.
Total variation distance: A metric measuring the largest difference in probabilities assigned by two probability measures. Total variation convergence is stronger and controls event probabilities uniformly.
Portmanteau theorem: A theorem giving equivalent characterizations of weak convergence of probability measures. It connects distributional convergence to inequalities for closed and open sets.
Convergence in mean: Convergence measured by the expected magnitude of the difference between random variables. Mean convergence can imply convergence in probability under suitable conditions, unlike distributional convergence alone.

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