Is the following interchange of limit and expectation correct?

Question

Let $(X_n)_n$ and $(Y_n)_n$ be infinite sequences of (possibly dependent) random variables in $\mathbb{R}_+$ and $f_n:\mathbb{R}_+^{2n}\to \mathbb{R}_+$. The sequences $(X_n)_n$ and $(Y_n)_n$ may be also dependent of each other. Assume also that each $X_n$ can take a finite number of values. Is the inequality $$ \liminf_{n\to\infty} \mathbb{E} \left[ f_n(X_1,\ldots,X_n, Y_1,\ldots,Y_n) \right] \ge \mathbb{E}_{X} \left[ \liminf_{N\to\infty} \mathbb{E}_{Y} \left[ f_n(X_1,\ldots,X_n, Y_1,\ldots,Y_n) \right]\right] $$ true where $\mathbb{E}_{X}$ and $\mathbb{E}_{Y}$ denote expectation with respect to $(X_n)_n$ and $(Y_n)_n$?

It sounds like Fatou's lemma. But I'm not sure that it can be applied if the sequences are dependent.

UPDATE. The inequality above perhaps should be written rigorously. An attempt: $$ \liminf_{n\to\infty} \mathbb{E} \left[ f_n(X_1,\ldots,X_n, Y_1,\ldots,Y_n) \right] = \liminf_{n\to\infty} \mathbb{E} \left[ \mathbb{E} [f_n(X_1,\ldots,X_n, Y_1,\ldots,Y_n) |(X_n)_n] \right] \ge \mathbb{E} \left[ \liminf_{N\to\infty} \mathbb{E} \left[ f_n(X_1,\ldots,X_n, Y_1,\ldots,Y_n) | (X_n)_n \right]\right] $$ This should be a direct application of Fatou's lemma but I am not sure. I'm not an expert of conditional expectations.

Answer 1

In your case you have to make sure that the random variables $\mathbb{E} [f_n(X_1,\ldots,X_n, Y_1,\ldots,Y_n) |(X_n)_n]$ are bounded from below, for all $n\in\mathbb{N}$, by a random varibale $\eta$ for which $\mathbb{E}[\eta]>-\infty$ but this already follows since $f$ is a positive function and thus the conditional expectations are positive random variables (you can take $\eta=0$). Hence you can apply Fatou .. and you obtain your inequality.