Hello everybody. I'm working about asymptotic estimates of
$M_n = \left|\sum_{k=1}^n Z_k\right|$
where $Z_1, Z_2, \ldots$ are independent uniformly distributed random variables on the complex unit circle. I found that the expectation $\textbf{E}[M_n^2] = n$ and the variance $\textbf{Var}[M_n^2] = n^2 – n$, so with Chebyshev's inequality I concluded that $M_n = o(n)$ almost surely as $n \to \infty$.
It is possible to improve this estimate? I need something like $M_n = O(\sqrt{n})$ or $M_n = O(n^{1/2 + \varepsilon})$. Thanks.
Best Answer
This is a classical random walk in the plane, extensively studied wrt Brownian motion etc.. Other people here are more expert in the subject than I am, so I'll leave it for them to provide you with references.
Also, because the summands are bounded, you can apply Hoeffding's inequality or similar to find strong tail bounds.