30 - Convergence of probability measures

Summary

Suppose we have a sequence of probabilities on a metric space S and we want to define what it means for the sequence to converge weakly. Alternately, we may have a sequence of random variables and want to say what it means for the random variables to converge weakly. We will apply the results we obtain here in later chapters to the case where S is a function space such as C[0, 1] and obtain theorems on the convergence of stochastic processes.

For now our state space is assumed to be an arbitrary metric space, although we will soon add additional assumptions on S. We use the Borelσ-field on S, which is the σ-field generated by the open sets in S.We write A⁰, Ā, and δA for the interior, closure, and boundary of A, respectively.

The portmanteau theorem

Clearly the de?nition of weak convergence of real-valued random variables in terms of distribution functions (see Section A. 12) has no obvious analog. The appropriate generalization is the following; cf. Proposition A. 41.

Definition 30.1 A sequence of probabilities {ℙ_n} on a metric space S furnished with the Borel σ-field is said to converge weakly to ℙ if ʃ f dℙ_n → ʃ fdℙ for every bounded and continuous function f on S. A sequence of random variables {X_n} taking values in S converges weakly to a random variable X taking values in S if E f (X_n) → Ef(X) whenever f is a bounded and continuous function.