I'm interested in the convergence of the tails of sums of probability distributions. For instance, let $$ X \sim \exp(-|x|^\delta) $$
and $Y_n = \sum_{i=0}^n X_i$, where $X_i$ are i.i.d. as $X$. What is the asymptotic decay of $Y_n$ as $x\rightarrow\infty$ ? For $\delta=1$, this can be done by explicitely computing $Y$:
$$ Y_n \sim \frac{2^{1/2-n} |x|^{n - 1/2} K_{1/2-n}(|x|)}{\sqrt{\pi} \Gamma(z)} $$
Combined with an asymptotic expansion, this yields $Y_n \sim \exp(-x)$ as $x\rightarrow\infty$. The central limit theorem states that as $n\rightarrow\infty$, $Y$ converges to a normal distribution. The speed of convergence is not uniform. Here lies my question: does the asymptotic behavior of unbounded distributions converges at all ? In other words, should I always expect that $Y_n \sim \exp(-x^\delta)$ as $x\rightarrow\infty$ and is this proven ?