Manuscript En01-C@1 · En01-C.tex · section · 00a2161c

Tail and implementation control

A bounded complete-record statistical certificate › Tail and implementation control · lines 1615–1657

En01-C@1#eq:clipping, En01-C@1#eq:Brecord

The mean signature involves an unbounded record. It cannot be inserted directly into Hoeffding's inequality. For $L > 0$ , let

c_{L} (z) = clip (z, - L, L) .

If $E_{h} Z_{i}^{2} \leq B_{h, i}$ , then

∣ E_{h} Z_{i} - E_{h} c_{L} (Z_{i}) ∣ \leq \frac{B _{h, i}}{L} .

Indeed, $∣ z - c_{L} (z) ∣ \leq z^{2} / L$ .

Put

B_{h}^{w} = i \sum ∣ a_{i} ∣ B_{h, i}, B = B_{0}^{w} + B_{1}^{w} .

These bounds must be uniform on the tested joint families.

A finite-sensor record bound is available directly from (eq:recordY). Writing $m_{t} = tr (x^{2} ρ_{c, t})$ ,

Z = \frac{2 ϵ _{m} μ}{τ} \int_{0}^{τ} m_{t} d t + \frac{W _{τ}}{2 τ} .

Using $(a + b)^{2} \leq 2 a^{2} + 2 b^{2}$ , Jensen's inequality, and $m_{t}^{2} \leq tr (x^{4} ρ_{c, t})$ , we obtain

E Z^{2} \leq 4 ϵ_{m} μ \int_{0}^{τ} ⟨ x^{4} ⟩_{t} d t + \frac{1}{2} .

No independence between the innovation and the conditional state was assumed in this estimate.

Displayed formulas in this passage

En01-C@1#eq:clipping · line 1624
$∣ E_{h} Z_{i} - E_{h} c_{L} (Z_{i}) ∣ \leq \frac{B _{h, i}}{L} .$
En01-C@1#eq:Brecord · line 1649
$E Z^{2} \leq 4 ϵ_{m} μ \int_{0}^{τ} ⟨ x^{4} ⟩_{t} d t + \frac{1}{2} .$
unlabeled · line 1620
$c_{L} (z) = clip (z, - L, L) .$
unlabeled · line 1632
$B_{h}^{w} = i \sum ∣ a_{i} ∣ B_{h, i}, B = B_{0}^{w} + B_{1}^{w} .$
unlabeled · line 1642
$Z = \frac{2 ϵ _{m} μ}{τ} \int_{0}^{τ} m_{t} d t + \frac{W _{τ}}{2 τ} .$