This problem is for a final exam I am taking in a graduate probability class. Collaboration has been explicitly allowed, but with the remark that the professor felt he couldn't stop us even if he wanted to.
The problem is this: Let $\alpha>0$ $X_1, X_2, ...$ be iid integer-valued random variables that have symmetric distribution about $0$ with $P(|X| \geq k)=1/k^\alpha \forall k \geq 1$. Let $S_n=\sum_{i=1}^n X_i$. For which $\alpha$'s is the random walk here recurrent?
I have already shown that when $\alpha>1$ then we have recurrence. I now want to show transience when that is not true, which is actually just a guess. It could well be that I'm mistaken. But I've been assuming I'm right, and according to some manipulations and Fourier analysis tricks, it comes down to showing that the following quantity is finite:
$Sup_{r \in (0, 1)}(2\pi)^{-1}*\int_{-2\pi}^{2\pi}1/(1-r\phi(t))dt<\infty$
Where $\phi(t)=\sum_{k=1}^\infty (1/k^\alpha-1/(1+k)^\alpha)(cos(kt))$
I am having trouble showing the above estimate, and it really comes down (if this solution is headed in the right direction) to just some inequality arguments on elementary functions and I just don't see it. I am trying to develop my senses in this area since complicated inequalities are a weakness of mine, so at first I would just like a fairly strong nudge. (I have been working on this problem for a week, it is the last one on my takehome exam.) I have already taken a few graduate courses in analysis (including the standard year-long core sequence) so things like that can be assumed. However, I know that there is a stronger statement that enables me to get away with just proving that the expression above with the sup over $r$ replaced by $r=1$ everywhere yields a finite number. But I don't know that result's proof, so please work with only the given expression unless you prove you can do otherwise.
I have tried various things like using small-angle approximations, but they aren't valid because the infinite sum makes it so that I have to be on smaller and smaller intervals and eventually such an approximation is only valid at $t=0$. I do not mean to discourage other sorts of answers, like full answers as opposed to hints, or of answers telling me to try an entirely new method not involving this sup of integrals. Anything is welcome and appreciated.
Edit: Some time has passed and I've fully given up. I'm ready for answers of any level of completeness.
This is example E2 in chapter 8 of Frank Spitzer's Principles of Random Walk (2nd edition). In fact, the walk is recurrent if and only if $\alpha\geq 1$. For $0<\alpha<2$, you can show that $1-\phi(t)\sim c\, |t|^\alpha$ as $t\to0$ for some $0<c<\infty$, and this gets you the result.