Asymptotic distribution of sequence of random variables.

Question

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Answer 1

You can rewrite $Z_n=\sum_{i=1}^n\frac{1_{L<X_i<R}}{n}$. Therefore by bilinearity $\text{Cov}(Y_n,Z_n)=\sum_{i,j=1}^n\frac{1}{n^2}\text{Cov}(X_i1_{L<X_i<R},1_{L<X_j<R})=\sum_{i=1}^n\frac{1}{n^2}\text{Cov}(X_i1_{L<X_i<R},1_{L<X_i<R})=\sum_{i=1}^n\frac{1}{n^2}\mathbb{E}[X_i1_{L<X_i<R}](1-\mathbb{P}[L<X_i<R])=\frac{1}{n}\mathbb{E}[X1_{L<X<R}](1-\mathbb{P}[L<X<R])$, and hopefully $F$ is nice enough that the expectation and probability are easy enough to calculate.

To answer your second question, not in the case of $Y_n$ and $Z_n$, as in general neither are normal and they may have quite complicated distributions. In the case of $Y^*$ and $Z^*$, you actually want to scale by $\frac{1}{\sqrt{n}}$, not $\frac{1}{n}$ to get a non-degenerate distribution, but if you do then by the multivariate Central Limit Theorem $(Y_n,Z_n)$ converge to $(Y^*,Z^*)$ where $(Y^*,Z^*)$ is a multivariate normal. You can easily find the mean and covariance matrix of this vector as the limit of the means, variances and covariances of $Y_n$,$Z_n$.