Theorem: Simultaneous sequence-mean certificate

Condition on fixed checkpoints independent of the fresh assessment sequences. For $L$ bounded summaries, each averaged over $n$ independent sequence blocks,

is a simultaneous mean radius at confidence at least $1-\delta$.

Within-sequence dependence and within-sequence pairing between checkpoints are allowed. A paired sequence contrast in an interval of length $B$ has radius

Proof. For one bounded mean, Hoeffding's inequality gives $2e^{-2n{r}^2}$ Hoeffding. Union bound over $L$ summaries. Rescale an interval of length $B$ to $[0,1]$. The independence is between sequence blocks, not their steps.