Repeated Hypothesis Testing on a Growing Data Set

In many problems, especially when data are collected over a long period of time, hypothesis testing is done repeatedly as new data arrive. It is shown both for one particular problem and for a more general class of problems that if testing is performed each time on the total amount of data accumulated, the true simple null hypothesis will be rejected at least once as the number of tests approaches infinity. Furthermore, it is conjectured that the conclusion holds for most problems of interest in which the null hypothesis is simple.