Manuscript Psy03-B@1 · Psy03-B-v1.tex · proposition · 8ca3caf9

Proposition: Normalization cost on rare histories

Enlarged Markov specification and reduced memory › Reference-stable error accounting · lines 2097–2121 · proof included or linked

Psy03-B@1#prop:normalization

For positive subnormalized blocks $τ, τ^{'}$ with traces $p, p^{'} \geq p_{m i n} > 0$ , if

\frac{1}{2} \norm τ - τ^{'}_{1} \leq e,

then

d_{tr} (τ / p, τ^{'} / p^{'}) \leq min {1, 2 e / p_{m i n}} .

Proof. Insert $τ^{'} / p$ , use $\abs p - p^{'} \leq \norm τ - τ^{'}_{1}$ , and obtain

\norm τ / p - τ^{'} / p^{'}_{1} \leq \frac{\norm τ - τ ^{'} _{1} + \abs p - p ^{'}}{p} \leq \frac{2 \norm τ - τ ^{'} _{1}}{p _{m i n}} .

Divide by two.

Displayed formulas in this passage

unlabeled · line 2101
$\frac{1}{2} \norm τ - τ^{'}_{1} \leq e,$
unlabeled · line 2105
$d_{tr} (τ / p, τ^{'} / p^{'}) \leq min {1, 2 e / p_{m i n}} .$
unlabeled · line 2114
$\norm τ / p - τ^{'} / p^{'}_{1} \leq \frac{\norm τ - τ ^{'} _{1} + \abs p - p ^{'}}{p} \leq \frac{2 \norm τ - τ ^{'} _{1}}{p _{m i n}} .$