On asymptotic correlation coefficient for some order statistics

By the relationship between a continuous population $X$ and the uniform distribution $U[0,1]$, we gain for a sample quantile an equivalent expression of its variance and for two different sample quantiles the asymptotic correlation coefficient. As the population of interest can have no expectation, the obtained conclusions are applicable to the location estimating problem of a Cauchy distribution. On that occasion, we finally obtained a quick and effective estimator established by a linear function of some sample quantiles. For similar problems, the presented approach is worthy of reference.