On probabilistic properties of conditional medians and quantiles

Conditional medians and quantiles are frequently used in analyzing time series data with heavy tails for their robustness properties. Most of the available literature focusses on statistical estimation of conditional quantiles and large sample behavior of the proposed estimators, whereas probabilistic properties of the true conditional medians themselves have not been fully explored (Tomkins [1975. On conditional medians. Ann. Probab. 3, 375-379; 1978. Convergence properties of conditional medians. Canad. J. Statist. 6, 169-177]). It is well-known that not all properties of conditional expectations have analogues for conditional medians. In this short note, we study to what extent analogues of certain properties of conditional expectations hold for conditional medians and generalize some of these properties to any general conditional quantile.