Theorem: Bounded complete-record test

If actual null scores have mean at most $a$, alternative scores have mean at least $b$, and $g=b-a>0$, then $L$ independent attempted runs with a midpoint threshold have

Thus

controls each error by ${\alpha }_{\ast }$.

The same bound holds for dependent runs if the corresponding mean bounds hold conditional on every past record. Unconditional mean bounds alone are insufficient.

Proof. For independence, apply Hoeffding's inequality at distance $g/2$ Hoeffding. For dependence, the conditional bounded-variable exponential bound is

Iterate and optimize. Repeating one common Bernoulli outcome shows the failure under arbitrary dependence.