| import OAI.Probability.GammaPassage.MeanLimit |
| import Mathlib.MeasureTheory.OuterMeasure.BorelCantelli |
| import Mathlib.Algebra.Order.Archimedean.Basic |
|
|
| namespace OAI |
|
|
| namespace GammaFPP |
| open MeasureTheory Filter Topology PlanarFPP |
|
|
| variable {Ω : Type*} [MeasurableSpace Ω] {P : Measure Ω} |
|
|
| / |
| almost-sure convergence to zero. The events need not be measurable. -/ |
| theorem ae_tendsto_zero_of_summable_deviations (X : ℕ → Ω → ℝ) |
| (hsum : ∀ k : ℕ, |
| (∑' n, P {ω | 1 / ((k : ℝ) + 1) ≤ |X n ω|}) ≠ ⊤) : |
| ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 0) := by |
| have hall : ∀ᵐ ω ∂P, ∀ k : ℕ, ∀ᶠ n in atTop, |
| ω ∉ {ω | 1 / ((k : ℝ) + 1) ≤ |X n ω|} := |
| ae_all_iff.mpr (fun k => ae_eventually_notMem (hsum k)) |
| filter_upwards [hall] with ω hω |
| apply Metric.tendsto_nhds.mpr |
| intro ε hε |
| obtain ⟨k, hk⟩ := exists_nat_one_div_lt hε |
| filter_upwards [hω k] with n hn |
| have hn' : |X n ω| < 1 / ((k : ℝ) + 1) := lt_of_not_ge hn |
| simpa only [Real.dist_eq, sub_zero] using hn'.trans hk |
|
|
| / |
| theorem ae_tendsto_of_summable_centered_deviations |
| (X : ℕ → Ω → ℝ) (a : ℕ → ℝ) (L : ℝ) |
| (ha : Tendsto a atTop (𝓝 L)) |
| (hsum : ∀ k : ℕ, |
| (∑' n, P {ω | 1 / ((k : ℝ) + 1) ≤ |X n ω - a n|}) ≠ ⊤) : |
| ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 L) := by |
| filter_upwards [ae_tendsto_zero_of_summable_deviations |
| (fun n ω => X n ω - a n) hsum] with ω hω |
| simpa only [sub_add_cancel, zero_add] using hω.add ha |
|
|
| / |
| limit with the Fekete limit of the means. -/ |
| theorem GammaEnvironment.ae_lattice_limit_of_summable_deviations |
| {τ : Edge → Ω → ℝ} {shape rate : ℝ} |
| (h : GammaEnvironment shape rate P τ) (hs : 0 < shape) (hr : 0 < rate) |
| (z : Vertex) |
| (hsum : ∀ k : ℕ, (∑' n : ℕ, P {ω | |
| 1 / ((k : ℝ) + 1) ≤ |
| |passageTime (fun e => τ e ω) 0 (n • z) / (n : ℝ) - |
| meanPassageSequence P τ z n / (n : ℝ)|}) ≠ ⊤) : |
| ∀ᵐ ω ∂P, Tendsto |
| (fun n : ℕ => passageTime (fun e => τ e ω) 0 (n • z) / (n : ℝ)) |
| atTop (𝓝 (directionalMeanConstant h hs hr z)) := |
| ae_tendsto_of_summable_centered_deviations _ _ _ |
| (h.meanPassageSequence_tendsto hs hr z) hsum |
|
|
| end GammaFPP |
|
|
| end OAI |
|
|
