Theorem: Restricted pattern summaries

For a common fixed horizon $H$, the event indicator ${\delta }_H$, the normalized restricted time $R_H/H$, and each indicator $1\{{\tau }_H\le h\}$, $h\le H$, are bounded sequence-level summaries. Their population means therefore admit Theorem thm:sequencehoeffding.

If a sequence contains several eligible switches, first use a predeclared bounded within-sequence aggregation. Those switches are not automatically independent replicates.

A censored record and an event completing exactly at $H$ have the same $R_H$, but different ${\delta }_H$. The censoring indicator must therefore be retained.

Proof. Each indicator is in $[0,1]$, and $0\le R_H/H\le 1$. Apply the sequence-level theorem. The last statement follows directly from the definition.