Limit of the Expectation of Monotonically Decreasing Sequence of Non-Negative R.V.s

Question

The Monotone Convergence Theorem states: Let $(X_n)_n$ be a sequence of random variables which are non-negative and increasingly converge to a random variable $X$, meaning that $0 \leq X_1 \leq X_2 \leq ... \leq X$ with $\lim_{n \rightarrow \infty} X_n = X$. Then, we can interchange limit with expectations: $$\lim_{n\rightarrow \infty} EX_n = E[\lim_{n \rightarrow \infty} X_n ] = EX.$$ I would wish to inquire under what condition(s) this 'interchange' could occur if instead given a sequence of random variables $(X_n)_n$ which are non-negative and decreasingly converge to $X$.

Answer 1

It would be enough for $EX_1 < \infty$. Indeed, if $X_n$ decrease to $X$, then since the sequence of random variables $X_1 - X_n$ are positive and increasing to $X_1 - X$, the MCT guarantees that $$E(X_1 - X_n) \to E(X_1 - X),$$ and since $0 \leq EX \leq EX_n \leq EX_1 < \infty$ one can now use linearity to recover $EX_n \to EX$.