For a broad class of jackknife statistics, it is shown that the Tukey estimator of the variance converges almost surely to its population counterpart. Moreover, the usual invariance principles (relating to the Wiener process approximations) usually filter through jackknifing under no extra regularity conditions. These results are then incorporated in providing a bounded-length (sequential) confidence interval and a preassigned-strength sequential test for a suitable parameter based on jackknife estimators.
"Some Invariance Principles Relating to Jackknifing and Their Role in Sequential Analysis." Ann. Statist. 5 (2) 316 - 329, March, 1977. https://doi.org/10.1214/aos/1176343797