Consider a 2D random walk with the magnitude of the nth step fixed by the function $f(n) = \frac{1}{n}$ and the direction being random.
I know that the root mean square comes out to be,
\begin{equation} \sqrt{\langle x^2\rangle} = \sqrt{\sum_{n=1} ^{\infty} \frac{1}{n^2}} = \frac{\pi}{\sqrt{6}} \end{equation}
So, the random walk has a finite width in which the particle will typically be.
But I am interested in understanding how quickly does this probability function decay off for values beyond this width. Does it decay off as a gaussian $e^{-a x^2}$, or much much faster than that? Is this problem solved somewhere or easy to solve? Any reference or help would be much appreciated.
If this problem is difficult to solve in it full generality, then can the rate of the decay of the probability function be somehow guessed?