33 Convergence of random variables
Convergence of expected values
Suppose \(X_n \to X\) almost surely. Then, each of the following is a sufficient condition for \(\EXP[X_n] \to \EXP[X]\):
Monotone Convergence Theorem. \(0 \le X_1 \le X_2 \cdots\).
Bounded Convergence Theorem. there exists a constant \(b\) such that \(|X_n| \le b\) for all \(n\).
Dominated Convergence Theorem. there exists a random variable \(Y\) such that \(|X_n| \le Y\) almost surely for all \(n\) and \(\EXP[Y] < ∞\).
\(\{X_n\}_{n \ge 1}\) is uniformly integrable.