Firstly, a well-known upper estimate concerning the distance of independent products of probability measures is extended to the case of signed measures. The upper bound depends on the total variation of the signed measures and on the distances of the single components where the distances are measured in the sup-metric. Under certain regularity conditions, the upper estimate can be sharpened by using asymptotic expansions. These expansions hold true over the set of all integrable function. An application of these results together with an asymptotic expansion of the distribution of a single order statistic yields an asymptotic expansion of the joint distribution of order statistics under the exponential distribution.
"Approximation of Product Measures with an Application to Order Statistics." Ann. Probab. 9 (2) 335 - 341, April, 1981. https://doi.org/10.1214/aop/1176994477