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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
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Minimal distance between random points on the unit circle
Fix $n$. Take the integers from $0$ to $n-1$ and define the distance between $x, y \in [0, n-1] \cap \mathbb{Z}$ as $d(x,y)=\min(|x-y|, n- |x-y|)$.
Now take $2k$ distinct points $x_1, \dots, x_{2k}$ …