Applying CLT to random variable made up of two sequences of iid random variables

2nd year stats hw

Q: Suppose you have a sequence $$X_1, X_2, ...$$ of iid random variables with mean $$E(X_1)=\mu_X$$ and variance $$Var(X_1)=\sigma^2_X$$ and another sequence $$Y_1, Y_2, ...$$ of iid random variables with mean $$E(Y_1)=\mu_Y$$ and variance $$Var(Y_1)=\sigma^2_Y$$. For each $$n=1,2,...$$ let $$A_n$$ be the random variable $$\frac{\sqrt n}{\sqrt {\sigma^2_X+\sigma^2_Y}}[\bar X_n - \bar Y_n - (\mu_X - \mu_Y)]$$ where $$\bar X_n = \sum_{i=1}^n \frac{X_i}{n}$$ and $$\bar Y_n = \sum_{i=1}^n \frac{Y_i}{n}$$.

Show that, in distribution, $$A_n$$ converges to $$N(0,1)$$ as $$n \to \infty$$.

I know that this will require use of the central limit theorem and when I asked my lecturer for help he just reminded me that the $$X$$ variables are independent to the $$Y$$ variables, but I don't know how to apply this. Please help - even if its just pointing me in the right direction!

• Independence of $(X_i)$ and $(Y_i)$ (from each other) has to be in the hypothesis. Othrwise the result is false. Commented Aug 30, 2020 at 0:16
• @BrianMoehring Ah yes I meant $(\mu_X - \mu_Y)$. Good pick up! Commented Aug 30, 2020 at 0:39
• @KaviRamaMurthy Yeah that's what I thought and so I am a bit confused why he reminded me that - it doesn't seem much help. Commented Aug 30, 2020 at 0:41

Defining $$Z_n = X_n - Y_n$$, by linearity of expectation $$E[Z_n] = E[X_n] - E[Y_n] = \mu_x - \mu_y$$. Using the properties of variance, we also have $$Var(Z_n) = E[(X_n - Y_n)^2] - E[X_n-Y_n]^2 = E[{X_n}^2] - 2 E[X_n Y_n] + E[{Y_n}^2] - (E[X_n]^2 - 2E[X_n]E[Y_n] + E[Y_n]^2) = (E[{X_n}^2] - E[X_n]^2) + (E[{Y_n}^2] - E[Y_n]^2) + 2(E[X_n]E[Y_n] - E[X_n Y_n]) = \sigma_x^2 + \sigma_y^2 + 2(E[X_n]E[Y_n] - E[X_n Y_n])$$ Because $$X_n$$ and $$Y_n$$ are independent, $$E[X_n Y_n] = E[X_n]E[Y_n]$$ so the variance simplifies to $$Var(Z_n) = \sigma_x^2 + \sigma_y^2$$. The sample mean of $$Z_n$$ is $$\frac{1}{n}\sum_{i} (X_i - Y_i) = \overline{X_n} - \overline{Y_n}$$, so $$A_n$$ can be rewritten as $$A_n = \frac{\sqrt{n}}{\sqrt{Var(Z_n)}}(\overline{Z_n} - E[Z_n])$$ which by the CLT converges in distribution to $$N(0,1)$$.
• What if $X_n$'s and $Y_n$'s are not independent? Commented May 20, 2022 at 11:44