# Random walks-heavy tailed case

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?