I have a question which I anticipated to be rather easy initially. After some googling, however, I realized it is actually not that easy. It concerns a 2-dimensional random walk with constant unit step-size, i.e., $$ Z=\sum_{k=1}^{M}e^{i(\phi_{k})} $$ where $\phi_{k}$ are independent and identically distributed random variables. Their probabilities are described by a uniform unit probability function between $0$ and $2\pi$ and zero elsewhere. If I'm correct, the expected root-mean-square distance is given by $\sqrt{M}$ for such a random walk. However, does there exist an analytical description of the probability of the root-mean-square distance. That is a probability density function of $|Z|$ as a function of $M$.
Thanks in advance!!


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