Convergence in distribution ($X_n\stackrel{d}{\to }X$)

Random variables $X_n$ converge in distribution, or converge weakly, to $X$ whenfor every bounded continuous function $f$. For real random variables this is equivalent to convergence of the distribution functions at every continuity point of the limiting distribution function.

For convergence in distribution of random elements in a separable complete metric space, one can construct copies of the sequence and its limit on one probability space which converge almost surely. The copies preserve the individual laws; they need not preserve the original dependence between sequence elements. Suitable more general versions permit a separably supported limit in a metric space.

If all moments of real random variables converge to those of a moment-determinate limit law, then the variables converge to that law in the sense of convergence in distribution. Moment determinacy is essential; the standard normal distribution meets this condition.

Probability measures ${\mu }_n$ on a metric space converge weakly to $\mu$ when $\int fd{\mu }_n\to \int fd\mu$ for every bounded continuous real function $f$.

A scaling limit of random curves is a weak convergence of probability measures after spatial rescaling, usually modulo monotone reparameterization. Tightness specifies the topology and yields subsequential limits; identification of their laws gives a unique continuum model. SLE is a central family of such limiting curve laws.

On càdlàg paths on a compact time interval, convergence in this topology means that increasing continuous bijections of the time interval can be chosen converging uniformly to the identity, while the corresponding time-changed paths converge uniformly to the limit. At a continuous limiting path this reduces to uniform convergence. It is a standard topology for weak limits of accelerated discrete-time chains, whose paths are step functions.

This is the coarsest topology making all displayed bounded-continuous test integrals continuous. On a separable metric space it is metrized by the bounded-Lipschitz metric. This probability-measure topology should be distinguished from the weak topology of a Banach space.

The bounded-Lipschitz metric tests probability measures against a uniformly bounded Lipschitz class. It metrizes weak convergence of probability measures on separable metric spaces. With a complete compatible base metric, it is complete on the probability measures of a Polish space. Replacing the sum norm by the maximum norm changes the metric only by uniform constant factors.

On a Polish space, a family of probability measures is relatively compact for weak convergence exactly when it is tight. For finite measures, one additionally requires uniformly bounded total masses.

A weakly closed uniformly tight family of Borel probability measures on a Polish space is weakly compact. Embed the underlying space into a compact metric closure, take subsequential pushforward limits, and use the closed-set Portmanteau theorem on the common tightness compact sets. The limit retains full mass in the original space; inverse continuity of the probability pushforward embedding theorem then returns the weak limit.

On a compact metric space, every sequence of Borel probability measures has a subsequence with weak convergence of probability measures to a Borel probability measure. To see this, choose a countable uniformly dense subset of continuous functions, extract a diagonal subsequence of their bounded integrals, and extend the resulting positive normalized functional to all continuous functions. The Riesz representation theorem supplies its Borel probability measure. This is useful for constructing an invariant measure from orbit empirical measures.

For random variables supported in one fixed compact interval, convergence of every moment is equivalent to weak convergence. One direction tests monomials; the other follows by uniformly approximating continuous functions by polynomials.

Two probability measures supported on one compact interval are equal if all their moments are equal. Indeed, equality on monomials gives equality on polynomials, the Weierstrass approximation theorem extends it to every continuous function on the interval, and compactly supported continuous functions determine a finite Borel measure.

The Portmanteau theorem gives equivalent formulations of weak convergence, including inequalities on open and closed sets and convergence of expectations for every bounded measurable function whose discontinuity set has zero limiting probability.

If $X_n\stackrel{d}{\to }X$ and a measurable function $g$ is continuous at $X$ almost surely, then $g(X_n)\stackrel{d}{\to }g(X)$. The same principle holds for random elements of metric spaces.