Manuscript Em03-B@2 · Em03-B-v2.tex · proposition · c1a0977d

Proposition: Exact referenced record tail

The B1–B3 mathematical interface › Exact tails and mean bias · lines 636–670 · proof included or linked

Em03-B@2#prop:recordtail, Em03-B@2#eq:idealtails

For every normal input,

d_{tr} (Ω_{N A} (ρ), Ω_{\infty A} (ρ)) = ρ Pr (N_{h} > N) .

Consequently, for the geometric marginal,

p_{N} (k) E_{ideal} κ d_{tr} (Ω_{N A}, Ω_{\infty A}) = (1 - q_{h}) q_{h}^{k} = ν_{h} (1 - q_{h}^{N}), = d_{TV} (p_{N}, p_{h}) = q_{h}^{N + 1} . B_{N} (ρ_{h}) (k < N), = ν_{h} q_{h}^{N}, p_{N} (N) = q_{h}^{N},

Proof. At record $N$ , the difference is the positive block $\sum_{n > N} ρ_{A}^{(n)}$ ; at each larger record it is $- ρ_{A}^{(n)}$ . Orthogonal record blocks have additive trace norm. The half trace norm is therefore the discarded probability. Equation (eq:generalbias) follows from $N_{h} - min (N_{h}, N) = (N_{h} - N)_{+}$ . Geometric summation gives $Pr (N_{h} \geq j) = q_{h}^{j}$ and

B_{N} = \frac{q _{h} ^{N + 1}}{1 - q _{h}} = ν_{h} q_{h}^{N} .

Displayed formulas in this passage

Em03-B@2#eq:idealtails · line 644
$p_{N} (k) E_{ideal} κ d_{tr} (Ω_{N A}, Ω_{\infty A}) = (1 - q_{h}) q_{h}^{k} = ν_{h} (1 - q_{h}^{N}), = d_{TV} (p_{N}, p_{h}) = q_{h}^{N + 1} . B_{N} (ρ_{h}) (k < N), = ν_{h} q_{h}^{N}, p_{N} (N) = q_{h}^{N},$
unlabeled · line 639
$d_{tr} (Ω_{N A} (ρ), Ω_{\infty A} (ρ)) = ρ Pr (N_{h} > N) .$
unlabeled · line 667
$B_{N} = \frac{q _{h} ^{N + 1}}{1 - q _{h}} = ν_{h} q_{h}^{N} .$

Proposition: Exact referenced record tail

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