Simultaneous domination for the positive-rate errors

References

• J. Bertoin, Lévy Processes, Cambridge University Press, 1996, Chapter IV.

theorem BGW.StableTrees.ae_abs_discountedErrorMartingale_le {θ : ℝ} (hθ : 1 < θ ∧ θ ≤ 2) {ε : ℝ} (hε : 0 < ε) (K : ℝ) (T : NNReal) : ∃ (q₀ : ℝ) (C : ℝ), 0 < q₀ ∧ ∀ᵐ (f : Skorokhod.DLoc) ∂↑(stableLocalLaw θ), ∀ s ≤ T, supReflectedFrom 0 f s ≤ K → ∀ (q : ℝ), 0 < q → q < q₀ → |discountedErrorMartingale θ q ε s f| ≤ ↑T / ε + supAuxLocalTime θ f T + C

Domination holds on one event for all times and all sufficiently small positive rates.