32  Convergence of random variables

Author

Affiliation

Aditya Mahajan

McGill University

Convergence of expected values

Suppose \(X_n \to X\) almost surely. Then, each of the following is a sufficient condition for \(\EXP[X_n] \to \EXP[X]\):

1. Monotone Convergence Theorem. \(0 \le X_1 \le X_2 \cdots\).

2. Bounded Convergence Theorem. there exists a constant \(b\) such that \(|X_n| \le b\) for all \(n\).

3. Dominated Convergence Theorem. there exists a random variable \(Y\) such that \(|X_n| \le Y\) almost surely for all \(n\) and \(\EXP[Y] < ∞\).

4. \(\{X_n\}_{n \ge 1}\) is uniformly integrable.