On the convergence of bivariate order statistics: Almost sure convergence and convergence rate

We study the convergence of bivariate order statistics, and get the almost sure convergence and convergence rate, generalizing one of main results in Huang et al. (2013). To be more precise, we prove the almost sure convergence of bivariate order statistics without the positive quadrant dependent (PQD) condition mentioned in Huang et al. (2013), and extend this result to a more general case. We also give the bound on the Kolmogorov distance between the distributions of bivariate order statistics and its limit. Our results provide an approximate algorithm for sampling from complicated structures, verified by some examples in the last section of this paper.