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Let $\beta > 0$ and $S_{0}=0$, and let $S_{n}=\xi_{1}+\dots+\xi_{n}$,$n \geq 1$, be a random walk with i.i.d. increments $\{\xi_{n}\}$ having a common distribution

$P(\xi_{1}=-1)=1-C_{\beta}$ and $P(\xi_{1}>t)=C_{\beta}e^{-t^{\beta}}$, $t \geq 0$,

where $C_{\beta} \in (0,1)$ is s.t. $E\xi_{1}=-1/2$. Let $M= \sup_{n \geq 0}S_{n}$.

Now, the question is for which values of $\beta > 0$ is it that the main reason for $M$ to be large is that there is a single large summand $\xi_{n}$ for some $n$? There is a hint: one has to identify the range of $\beta$ for which the distribution of $\xi_{1}$ is heavy-tailed. First, I tried to understand why 'answering' the hint answers the main question. So if we show that the distribution of $\xi_{1}$ is heavy-tailed (for some $\beta$'s) then for those values of $\beta$, $\xi_{2},\dots,\xi_{n}$ come from the same heavy-tailed distribution. Thus for some $n$, one of the $\xi_{n}$'s will be somehow extreme, i.e. large, causing the sum to be large. Is that logic correct?

I know that, generally speaking, heavy-tailed distributions are probability distributions whose tails are not exponentially bounded. Still, I'm not really sure what I have to do in order to show for which $\beta$ the distribution of $\xi_{1}$ is heavy-tailed. In general, how does one show that a distribution is heavy-tailed?

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