Suppose $X_1, X_2, ..., X_n$ are sequences of iid random variables with mean equal to zero and variance = $σ^ 2$. we define $Y_n = \frac{s_n}{σ∗\sqrt{n}} −\frac{s_{2n}}{σ∗\sqrt{2n}}$ , $S_n = X_1 + X_2 +...+ X_n$ Using the central limit theorem, obtain the limit of the sequence $Y_n$ when n → ∞.
I tried a lot but I really don't know how to solve this problem, Can anybody help me pls?
I guess that you know the most classical version of the central limit theorem for i.i.d. sequences. It is possible to express $Y_n$ as a partial sum of independent random variables, but it will not be a sum of i.i.d.
To overcome the problem, one can write $$ Y_n=Z_n+Z'_n, Z_n=\frac{1}{\sigma \sqrt n}\left(1-\frac{1}{\sqrt 2}\right)S_n, Z'_n=-\frac{1}{\sigma \sqrt n}\left(S_{2n}-S_n\right). $$ The random variables $Z_n$ and $Z'_n$ are independent hence the characteristic function of $Z_n$ is the product of the characteristic function of $Z_n$ with that of $Z'_n$. The limit of these ones can be computed by the classical central limit theorem and it will give you the limiting distribution of $Y_n$.