# Sequential Resetting Procedure

The authors propose sequential resetting procedures built on e-values and test supermartingales that reset the capital to one after each rejection and continue testing, thereby reporting multiple intervals containing null violations; a refinement rule reports only observations after the within-block minimum of the capital. Under independent null e-values the FDR bound is α, while under dependence between null and non-null data the bound is α times a logarithmic factor with anytime validity, demonstrated on LLM watermark detection and ES backtesting.

The authors propose sequential resetting procedures built on e-values and test supermartingales that reset the capital to one after each rejection and continue testing, thereby reporting multiple intervals containing null violations; a refinement rule reports only observations after the within-block minimum of the capital. Under independent null e-values the FDR bound is α, while under dependence between null and non-null data the bound is α times a logarithmic factor with anytime validity, demonstrated on LLM watermark detection and ES backtesting.