Lemma. Prefix-window exposure is uniformly capped [ftip-00DK]
Lemma. Prefix-window exposure is uniformly capped [ftip-00DK]
Let \(p,w\in \mathbb N_0\) and let \(\kappa ^{p,w}\) be the prefix-window policy of Definition [ftip-00DD]. For every history and every \(t\),
\[ \left |\kappa ^{p,w}_t(h_t)\right |\leq p+w. \]Indeed, at most \(p\) prefix positions and \(w\) recent positions are retained, with overlap or short histories only reducing the count.