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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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Independence between $X_{n-k:n}$ and $\sum\limits_i Y_{n-i:n}-Y_{n-k:n}$
If $(X_i,Y_i), i=1,\ldots,n,$ is i.i.d sample from the joint distribution $F$ and there is dependence between the two variables say $R$. Denote the order statistics for the two variables $X_{1:n},\ldo …