Manuscript Psy04-B@1 · Psy04-B-v1.tex · proposition · 62c2b6f0

Proposition: Explicit storage and detector input-stability allowance

Complete records and robust score allowances › Parameter stability · lines 1548–1589 · proof included or linked

Psy04-B@1#prop:parameterstability, Psy04-B@1#eq:parameterstability

Consider two specified protocols with the same number of storage steps and respective parameters

(p, v, ϵ, η_{1}, \dots, η_{T}), (p^{'}, v^{'}, ϵ^{'}, η_{1}^{'}, \dots, η_{T}^{'}) .

Under the independent-flag model,

∣ Q_{score} - Q_{score}^{'} ∣ \leq \frac{1}{2 2} (∣ p - p^{'} ∣ + ∣ v - v^{'} ∣ + 2∣ ϵ - ϵ^{'} ∣ + t = 1 \sum^{T} ∣ η_{t} - η_{t}^{'} ∣) .

In particular,

t \prod η_{t} - t \prod η_{t}^{'} \leq t \sum ∣ η_{t} - η_{t}^{'} ∣.

Proof. For factors $u_{ℓ}, u_{ℓ}^{'} \in [0, 1]$ , telescoping gives

ℓ \prod u_{ℓ} - ℓ \prod u_{ℓ}^{'} = ℓ \sum (k < ℓ \prod u_{k}^{'}) (u_{ℓ} - u_{ℓ}^{'}) (k > ℓ \prod u_{k}) .

Take absolute values and use factors at most one. Apply this identity to $p, v, 1 - 2 ϵ, η_{1}, \dots, η_{T}$ .

Displayed formulas in this passage

unlabeled · line 1552
$(p, v, ϵ, η_{1}, \dots, η_{T}), (p^{'}, v^{'}, ϵ^{'}, η_{1}^{'}, \dots, η_{T}^{'}) .$
Psy04-B@1#eq:parameterstability · line 1557
$∣ Q_{score} - Q_{score}^{'} ∣ \leq \frac{1}{2 2} (∣ p - p^{'} ∣ + ∣ v - v^{'} ∣ + 2∣ ϵ - ϵ^{'} ∣ + t = 1 \sum^{T} ∣ η_{t} - η_{t}^{'} ∣) .$
unlabeled · line 1568
$t \prod η_{t} - t \prod η_{t}^{'} \leq t \sum ∣ η_{t} - η_{t}^{'} ∣.$
unlabeled · line 1578
$ℓ \prod u_{ℓ} - ℓ \prod u_{ℓ}^{'} = ℓ \sum (k < ℓ \prod u_{k}^{'}) (u_{ℓ} - u_{ℓ}^{'}) (k > ℓ \prod u_{k}) .$