Random walk with decreasing steps

Question

I 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. I am interested in understanding the behavior of $R(t)$ for large $t$.

For example can we estimate 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 variance can grow with $t$. Any information regarding the behavior of $R$ is appreciated.

Answer 1

Let $a:=\alpha$. Note that \begin{equation} Ee^{zX_n}=\frac{\sinh(z/n^a)}{z/n^a} \end{equation} for real $z>0$. Using the inequality $\dfrac{\sinh u}u<e^{u^2/6}$ for real $u\ne0$ (see e.g. this MathSE answer) and the independence of the $X_n$'s, we get \begin{equation} Ee^{zR(t)}\le e^{z^2 B_{a,t}/6}, \end{equation} where \begin{equation} B_{a,t}:=\sum_{n<t}\frac1{n^{2a}}. \end{equation} So, for any real $x>0$, \begin{equation} P(R(t)\ge x)\le e^{-zx+z^2 B_{a,t}/6}. \end{equation} The latter bound on $P(R(t)\ge x)$ is minimized at $z=3x/B_{a,t}$. Thus, \begin{equation} P(R(t)\ge x)\le e^{-3x^2/(2B_{a,t})}. \end{equation}