Suppose that $X_1,X_2,\ldots,X_n$ are independent and identically distributed random variables having characteristic function $\chi(t)=e^{-|t|^{1.9}}$.Then what is the weak limit of $n^{-5/9}S_n$ as n gets large, where $S_n=\sum_{k=1}^n X_k$?
I was trying to express the CDF of Sn in terms of the common CDF of Xi's and then use the relation between CDF and characteristic functions via the inversion formula.
And what could possibly be said when 1.9 and 5/9 are replaced by any arbitrary exponent?
Can anyone help or suggest me anything? Thanks in advance.
HINT
Let $\chi_n(t)$ denote the characteristic function of $S_n$, then $$ \chi_n(t) = \chi(t)^n = \left(e^{-|t|^{1.9}}\right)^n = e^{-n|t|^{1.9}}. $$
Can you simplify this based on $t$? e.g. if $|t|>1$ or if $|t|<1$ etc.?