Weak convergence and the Portmanteau theorem

Let μn and μ be probability measures on a metric space, with its Borel sets. Weak convergence means that integrating any bounded continuous real function against μn approaches its integral against μ. Boundedness prevents a tiny amount of probability at an enormous value from dominating the test. Continuity prevents the test from magnifying an arbitrarily small movement across a sharp threshold. On the real line this is equivalent to convergence of cumulative distribution functions at every continuity point of the limiting distribution function.

For a deterministic point mass at 1/n, bounded continuous tests tend to their value at zero, so the laws converge weakly to a point mass at zero. Yet the probability of the singleton containing zero is always zero before the limit and one at the limit. Weak convergence did not fail; the indicator of that singleton is discontinuous at the place where the limit concentrates all its mass. Event probabilities therefore need a boundary condition.

The Portmanteau theorem states equivalent criteria for weak convergence. For every closed set, the upper limit of its probabilities is at most its limiting-law probability. For every open set, the lower limit is at least its limiting-law probability. A closed set can acquire mass when nearby points approach its boundary, whereas an open set can lose boundary mass. These one-sided inequalities are oriented differently for a reason; reversing them would already fail for moving point masses.

If a Borel event A has boundary of μ-measure zero, applying the two inequalities to its interior and closure traps its probabilities between two quantities tending to the same value. Then μn(A) tends to μ(A). Such an event is called a continuity set of the limiting measure. The condition is sufficient, not necessary for a particular sequence. On the real line, a threshold event is safe when the limiting law has no atom at the threshold.

A continuous map preserves weak convergence after pushing the measures forward. More generally, a measurable map works when its discontinuity set has zero probability under the limiting law. This is the continuous mapping theorem. Squaring a real random variable is continuous everywhere; thresholding it is discontinuous at the threshold. The distinction explains why a smooth change of coordinates is safe while a decision rule near a limiting atom needs special care.

Weak convergence alone does not control unbounded expectations. A variable equal to n² with probability 1/n and zero otherwise converges in probability, hence in distribution, to zero; its expectation is n. Additional tail control, often uniform integrability, is needed to pass such moments through a weak limit. Tightness is a different idea: it prevents probability mass escaping every compact region and provides convergent subsequences under suitable space assumptions. It does not alone bound the contribution of very large values to an expectation.