Continuous mapping theorem

The continuous mapping theorem states that continuous functions preserve convergence in distribution and, under suitable conditions, convergence in probability or almost surely.

Convergence in distribution: A mode of convergence in which random variables' probability distributions approach a limiting distribution. The theorem's central guarantee is that continuous transformations preserve this mode of convergence.
Delta method: An asymptotic method that approximates the distribution of a smooth function of an estimator. It combines the continuous mapping theorem with local linear approximation.
Discontinuous mapping theorem: A refinement that permits discontinuous transformations when their discontinuity sets have zero probability under the limit. It extends the theorem precisely where global continuity is unavailable.
Probability measure: A measure assigning probabilities to events in a measurable space. Distributional convergence compares probability measures induced by the random variables.
Convergence in probability: Random variables converge in probability when the chance of deviating from their limit by any fixed amount tends to zero. The theorem also transfers this stronger form of convergence through continuous functions.
Slutsky's lemma: A result allowing sums, products, and ratios of convergent random variables to inherit suitable limiting distributions. It is a standard tool for simplifying asymptotic expressions using continuous mappings.
Convergence in total variation: A strong mode of convergence of probability measures controlling probability differences over all measurable events. Unlike distributional convergence, it can control transformations that are not continuous.
Random variable: A measurable function from a probability space to a measurable value space. The theorem transforms random variables and studies the limits of their distributions.
Almost sure convergence: Random variables converge almost surely when they converge pointwise outside an event of probability zero. Pointwise continuity ensures almost sure limits pass through the function.
Weak convergence of random variables: Convergence of probability measures against bounded continuous test functions. The theorem is a basic operation rule for weakly convergent random variables.

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Skorokhod representation theoremRelated: A represented almost-sure limit can yield distributional limits after applying suitable functions.