Combining p-values via averaging

This note discusses the problem of multiple testing of a single hypothesis, with a standard goal of combining a number of p-values without making any assumptions about their dependence structure. An old result by Rueschendorf shows that the p-values can be combined by scaling up their arithmetic mean by a factor of 2 (but no smaller factor is sufficient in general). More recent results in mathematical finance show, in addition, that $K$ p-values can be combined by scaling up their geometric mean by a factor of $e$ and by scaling up their harmonic mean by a factor of $e\ln K$. This leads to a generalized version of the Bonferroni-Holm method.