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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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Martingale Approach for upper bound on time until n deaths
suppose we have $n$ Geometric$(p)$ random variables $X_1,\dots,X_n$, and let $Y_t$ be the number of these random variable still alive at time $t$, i.e.
$Y_t = \sum_{i=1}^n \mathbb{1}\{X_i \geq t\}$
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Coupling Marginals of Distributions on the Sphere
Given a distribution $P_X$ on $\mathbb{R}$, when does there exist a coupling (i.e. joint distribution) $P_{X^n}$ of $X_1,...,X_n$, each distributed according to $P_X$, such that $\sum X_i^2 = n$ almos …