Magnitude of the sum of complex i.u.d. random variables in the unit circle - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T09:03:13Z http://mathoverflow.net/feeds/question/89478 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89478/magnitude-of-the-sum-of-complex-i-u-d-random-variables-in-the-unit-circle Magnitude of the sum of complex i.u.d. random variables in the unit circle Richard Bonne 2012-02-25T11:23:36Z 2012-02-25T13:43:22Z <p>Hello everybody. I'm working about asymptotic estimates of</p> <p>$M_n = \left|\sum_{k=1}^n Z_k\right|$</p> <p>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 $n \to \infty$.</p> <p>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.</p> http://mathoverflow.net/questions/89478/magnitude-of-the-sum-of-complex-i-u-d-random-variables-in-the-unit-circle/89479#89479 Answer by Brendan McKay for Magnitude of the sum of complex i.u.d. random variables in the unit circle Brendan McKay 2012-02-25T11:41:17Z 2012-02-25T13:28:19Z <p>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.</p> <p>Also, because the summands are bounded, you can apply Hoeffding's inequality or similar to find strong tail bounds.</p> http://mathoverflow.net/questions/89478/magnitude-of-the-sum-of-complex-i-u-d-random-variables-in-the-unit-circle/89486#89486 Answer by Douglas Zare for Magnitude of the sum of complex i.u.d. random variables in the unit circle Douglas Zare 2012-02-25T13:43:22Z 2012-02-25T13:43:22Z <p>$O(\sqrt n)$ is too much to ask, and it fails almost surely. The <a href="http://en.wikipedia.org/wiki/Law_of_the_iterated_logarithm" rel="nofollow">law of the iterated logarithm</a> says the real and imaginary parts are a. s. $O(\sqrt{n \log \log n})$ so with probability $1$, $M_n$ is $O(\sqrt{n \log \log n})$.</p>