pr.probability - A probability exercise related to Central Limit Thm

The fact that $b_nto infty$ is quite easy to check: if not, there is a $M$ and a subsequence $(b_{n'})$ which remains below $M$, hence $frac 1{b_{n'}}max_{1leqslant jleqslant n'}|X_j|geqslant frac 1{M}max_{1leqslant jleqslant n'}|X_j|$, hence $max_{1leqslant jleqslant n'}|X_j|$ would converge to $0$ in probability, which is not possible.

Let us denote the convergence in distribution by $Rightarrow$.

Theorem 7.6 in Durrett's book Probability: theory and examples, (second edition), provides an useful theorem here:

The convergence of types theorem. Assume that $Y_nRightarrow Y$ and there are constants $alpha_n>0$, $beta_n$ such that $Y'_n=alpha_nY_n+beta_nRightarrow Y'$, where $Y$ and $Y'$ are not degenerate. Then there are $alpha>0$ and $betainBbb R$ such that $alpha_nto alpha$ and $beta_nto beta$.

The proof uses the fact that in case of convergence in distribution, the sequence of corresponding characteristic functions actually converges uniformly on compact sets. Then we proves that for $alpha$, there is an unique positive real number which can be a candidate, and the same for $beta_n$.

Here, we use this theorem with $W_n:=frac{S_n-a_n}{b_n}$, $alpha_n:=frac{b_n}{b_{n+1}}$ and $beta_n:=frac{a_n-a_{n+1}}{b_{n+1}}$. It works since $frac{X_{n+1}}{b_{n+1}}$ goes to $0$ in probability.

Finally, we have to use independence in order to identify the obtained limits. We compute the characteristic function of $W_n$, and of $W_{n+1}$, and we take the limits.