Manuscript Hawk02-B@2 · Hawk02-B.tex · lemma · 95ea744f

Lemma: Sequential channel comparison

Conventions and elementary channel estimates · lines 320–345 · proof included or linked

Hawk02-B@2#lem:telescoping

Suppose $Φ_{j}, Φ_{j}$ are channels on common retained spaces and $d_{⋄} (Φ_{j}, Φ_{j}) \leq η_{j}$ . Then

d_{⋄} (Φ_{m} \dots Φ_{1}, Φ_{m} \dots Φ_{1}) \leq min {1, j = 1 \sum^{m} η_{j}} .

Proof. Insert one channel replacement at a time:

Φ_{m} \dots Φ_{1} - Φ_{m} \dots Φ_{1} = j \sum Φ_{m} \dots Φ_{j + 1} (Φ_{j} - Φ_{j}) Φ_{j - 1} \dots Φ_{1} .

Contractivity and the triangle inequality prove the bound.

Displayed formulas in this passage

unlabeled · line 326
$d_{⋄} (Φ_{m} \dots Φ_{1}, Φ_{m} \dots Φ_{1}) \leq min {1, j = 1 \sum^{m} η_{j}} .$
unlabeled · line 335
$Φ_{m} \dots Φ_{1} - Φ_{m} \dots Φ_{1} = j \sum Φ_{m} \dots Φ_{j + 1} (Φ_{j} - Φ_{j}) Φ_{j - 1} \dots Φ_{1} .$