I we have a random walk $$R(t)= \sum_{n<T} X_n,$$ with $X_n \sim U(-\tfrac{1}{n^\alpha}, \tfrac{1}{n^\alpha}),$ where $X_n$ are independant and $\alpha >0$.

I think that someone must have studied this before. My question is that what is the PDF of $R$, and in particular what is the probability of $R(t) \in [1, x)$?

Obviously, $E(R(t))=0,$ and $Var(R)= \tfrac{1}{3}\sum_{n<t} \tfrac{1}{n^{2\alpha}}.$ Therefore depends on $\alpha,$ the varianc (and some of the higher moments) can grow with $t$ but for $k>1/\alpha,$ we have $$E(R^k(t)) \leq \text{ constant}.$$

Any information regarding the behaviour of $R$ is appreciated.