Sum Infinite Random Variables

Question

Let's say we generate $n$ samples independently from two independent distributions $X$ and $Y$. We know that the following is true from Jensen's Inequality: $$\ E\left[\min\left(\sum_{i=1}^{n}X_i, \sum_{i=1}^{n}Y_i\right)\right] \leq \min\left(\sum_{i=1}^{n}E[X_i], \sum_{i=1}^{n}E[Y_i]\right) $$ I was wondering what happens if $n \to \infty$. Precisely, $$\ \lim_{n\to \infty}E\left[\min\left(\sum_{i=1}^{n}X_i, \sum_{i=1}^{n}Y_i\right)\right] = \lim_{n\to \infty}E\left[n\times \min\left(\frac{\sum_{i=1}^{n}X_i}{n}, \frac{\sum_{i=1}^{n}Y_i}{n}\right)\right] $$ From Strong Law of Large Numbers we have, \begin{align} \lim_{n\to \infty}E\left[n\times \min\left(\frac{\sum_{i=1}^{n}X_i}{n}, \frac{\sum_{i=1}^{n}Y_i}{n}\right)\right]&=\lim_{n\to \infty}E[n\times \min(\mu_X, \mu_Y)]\\&=\lim_{n\to \infty}n\times \min(\mu_X, \mu_Y) \end{align} For the second term, $$\ \min\left(\sum_{i=1}^{n}E[X_i], \sum_{i=1}^{n}E[Y_i]\right)=n\times \min(\mu_X, \mu_Y) $$ I know that the above results don't mean much as we have $\lim_{n\to \infty}n\times constant$. However, my intuition says that for large values of $n$ both the terms should be pretty close. Can you please explain if my intuition is actually correct? Can you please give a proof for the same or a counter-example?

Answer 1

Example:

Let $X$ and $Y$ be independent Gaussian $N(0,1)$. Let $\{X_i\}_{i=1}^{\infty}$ and $\{Y_i\}_{i=1}^{\infty}$ be independent and i.i.d. Gaussian $N(0,1)$. Then $$ \min\left[\sum_{i=1}^nE[X_i], \sum_{j=1}^nE[Y_j]\right] = 0 \quad \forall n \in \{1, 2, 3, ...\}$$ However, $\sum_{i=1}^n X_i$ has the same distribution as $\sqrt{n}X$ and so for all $n \in \{1, 2, 3, ...\}$ we have by independence of $\{X_i\}$ and $\{Y_i\}$: \begin{align} E\left[\min\left[\sum_{i=1}^n X_i, \sum_{j=1}^n Y_j \right] \right] &= E\left[\min\left[\sqrt{n}X, \sqrt{n}Y\right]\right] \\ &=\sqrt{n}\underbrace{E[\min[X,Y]]}_{<0} \end{align} and so this goes to $-\infty$ as $n\rightarrow\infty$.

According to Wolfram-Alpha we have $$ E[\min[X,Y]] = \frac{-1}{\sqrt{\pi}} \approx -0.56419 $$

https://www.wolframalpha.com/input/?i=int+int+1%2F%282+pi%29+min%28x%2Cy%29exp%28-x%5E2%2F2%29exp%28-y%5E2%2F2%29%2C+x%3D-infty..infty%2C+y%3D-infty..infty