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Geometric parameters satisfy the logarithmic frozen stopping and growth conditions

Statement

Let $ν, c, α_{*}, τ > 0$ , $0 < θ < 1$ and $0 < q < 1$ . Suppose $H$ is the conjugate of a truncated logarithmic source index with $H (u) \sim C (\log u)^{- ν}$ as $u \to \infty$ . Starting sufficiently far down the geometric sequence, relabel it as $α_{n} = α_{*} θ^{n + n_{0}}$ , $n \geq 1$ , and put $h_{n} = H (c / α_{n})$ and $ρ_{n} = α_{n} h_{n}$ . Then one can choose $n_{0}$ and $C_{H} > 1$ , fixed independently of noise, such that $h_{n} \leq C_{H} h_{n + 1}, q C_{H} < 1, \frac{ρ_{n}}{ρ_{n + 1}} \leq M : = θ^{- 1} C_{H} (n \geq 1) .$ The positive sequence $ρ_{n}$ decreases strictly to zero. For any positive target $w ↓ 0$ , let $N (w)$ be the last index with $ρ_{N (w)} \geq τ w$ . For sufficiently small $w$ this index exists and is at least two, and $τ w \leq ρ_{N (w)} < M τ w, h_{N (w)} ≍ [\log (1 / w)]^{- ν}, α_{N (w)} ≍ w [\log (1 / w)]^{ν} .$ In particular $w = δ^{2}$ gives $h_{N} ≍ [\log (1 / δ)]^{- ν}$ and $α_{N} ≍ δ^{2} [\log (1 / δ)]^{ν}$ . The nominal endpoint target $w = δ^{2} [\log (1 / δ)]^{- ν}$ gives the same $h_{N}$ and $α_{N} ≍ δ^{2}$ . In that endpoint case, an actual noise bound $ε_{δ}^{2} = O (w)$ and tolerances $η_{n} \leq C_{η} ρ_{n}$ give exactly the noise/tolerance scale required for applying a frozen recursion at this stopping index.

Statement references

Logarithmic source indices and their admissible parameter scales

Kaltenbacher, Theorem 7

Facts & Assumptions

Given: The source index, fixed geometric sequence, contraction number and target in the Statement.

[F1]

The truncated logarithmic conjugate is positive and finite at every positive argument, is nonincreasing, and has the stated asymptotic. Logarithmic source indices and their admissible parameter scales

Proof

1.1F1givenchoosealgebra

Positivity follows directly by testing the supremum at a sufficiently small $s > 0$ : the logarithmic index has $ψ (s) / s \to \infty$ , so $ψ (s) - u s > 0$ for any fixed $u > 0$ . Finiteness and monotonicity follow from boundedness of $ψ$ and from its defining supremum. By [F1], $h_{n} \sim C [(n + n_{0}) ∣ \log θ ∣]^{- ν}$ , so $h_{n} / h_{n + 1} \to 1$ . Fix any $C_{H} \in (1, 1 / q)$ and choose $n_{0}$ large enough that all ratios in the relabeled tail are at most $C_{H}$ . Since $α_{n}$ decreases strictly and $h_{n}$ is positive and nonincreasing, $ρ_{n}$ decreases strictly; its asymptotic is a positive constant times $θ^{n + n_{0}} (n + n_{0})^{- ν}$ , hence it tends to zero and its consecutive ratio is at most $M$ .

2.1step 1.1givenalgebra

For sufficiently small $w$ , $ρ_{2} \geq τ w$ , and strict decrease to zero gives a unique last index $N = N (w) \geq 2$ . It satisfies $ρ_{N} \geq τ w > ρ_{N + 1}$ , so $ρ_{N} \leq M ρ_{N + 1} < M τ w$ . In particular $N \to \infty$ as $w \to 0$ . Taking logarithms in the geometric asymptotic gives $(N + n_{0}) ∣ \log θ ∣ + ν \log (N + n_{0}) = \log (1 / w) + O (1)$ . Since $\log N / N \to 0$ , it follows that $N \sim \log (1 / w) / ∣ \log θ ∣$ . Substituting in $h_{N}$ yields $h_{N} ≍ [\log (1 / w)]^{- ν}$ , and $α_{N} = ρ_{N} / h_{N}$ gives the asserted parameter scale.

3.1step 2.1givenalgebra∎

For $w = δ^{2}$ , $\log (1 / w) = 2 \log (1 / δ)$ , giving the first specialization. For $w = δ^{2} [\log (1 / δ)]^{- ν}$ , $\log (1 / w) = 2 \log (1 / δ) + ν \log \log (1 / δ) \sim 2 \log (1 / δ)$ , and the same formulas give $h_{N} ≍ [\log (1 / δ)]^{- ν}$ and $α_{N} ≍ δ^{2}$ . If $ε_{δ}^{2} \leq C w$ , use the positive noise upper bound $\sqrt{C w}$ , increasing $C$ if necessary. The stopping bracket then has fixed constants $τ / C$ and $M$ , while every earlier $ρ_{n} \geq ρ_{N} \geq τ w$ and the stated tolerance bound is already $η_{n} \leq C_{η} α_{n} h_{n}$ . Thus all parameter/noise/tolerance conditions hold at that upper-bound scale.

Depends on

Logarithmic source indices and their admissible parameter scales

Used by

Frozen Newton rates under differential range invariance Theorem

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Barbara Kaltenbacher, Convergence rates under a range invariance condition with application to electrical impedance tomography (standard reference, not scraped)