| import OAI.Probability.InvariantIsing.Pressure.AlmostSure |
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| namespace OAI |
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| /-! Eventual local exponential tails imply almost-sure convergence whenever |
| the localizing random bound is eventually finite. -/ |
| noncomputable section |
| open MeasureTheory Filter Set |
| open scoped Topology |
| namespace InvariantIsing |
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| lemma exponential_tail_extend_initial {Ω : Type*} [MeasurableSpace Ω] |
| (P : Measure Ω) [IsProbabilityMeasure P] (E : ℕ → Set Ω) |
| {C a : ℝ} (hC : 0 ≤ C) (ha : 0 < a) (N₀ : ℕ) |
| (hE : ∀ n, N₀ ≤ n → P.real (E n) ≤ C*Real.exp (-a*(n : ℝ))) : |
| ∃ D : ℝ, 0 ≤ D ∧ ∀ n, P.real (E n) ≤ D*Real.exp (-a*(n : ℝ)) := by |
| let D := C+Real.exp (a*(N₀ : ℝ)) |
| have hCD : C ≤ D := le_add_of_nonneg_right (Real.exp_pos _).le |
| refine ⟨D,by dsimp [D]; positivity,fun n => ?_⟩ |
| by_cases hn : N₀ ≤ n |
| · exact (hE n hn).trans (mul_le_mul_of_nonneg_right hCD (Real.exp_pos _).le) |
| · have hle : (n : ℝ) ≤ N₀ := by exact_mod_cast (show n ≤ N₀ by omega) |
| have he : 1 ≤ Real.exp (a*(N₀ : ℝ))*Real.exp (-a*(n : ℝ)) := by |
| rw [← Real.exp_add,Real.one_le_exp_iff] |
| nlinarith |
| have hED : Real.exp (a*(N₀ : ℝ)) ≤ D := by dsimp [D]; linarith |
| exact (measureReal_le_one : P.real (E n) ≤ 1).trans |
| (he.trans (mul_le_mul_of_nonneg_right hED (Real.exp_pos _).le)) |
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| lemma ae_tendsto_zero_of_localized_exponential_tail {Ω : Type*} [MeasurableSpace Ω] |
| (P : Measure Ω) [IsProbabilityMeasure P] (X Z : ℕ → Ω → ℝ) |
| (htail : ∀ K ε : ℝ, 0 < K → 0 < ε → |
| ∃ C a : ℝ, 0 ≤ C ∧ 0 < a ∧ ∃ N₀ : ℕ, |
| ∀ n, N₀ ≤ n → P.real {ω | ε ≤ |X n ω| ∧ Z n ω ≤ K} ≤ C*Real.exp (-a*(n : ℝ))) |
| (hbound : ∀ᵐ ω ∂P, ∃ K : ℝ, ∀ᶠ n in atTop, Z n ω ≤ K) : |
| ∀ᵐ ω ∂P, Tendsto (fun n => X n ω) atTop (𝓝 0) := by |
| let T := fun k n ω => if Z n ω ≤ (k+1 : ℝ) then X n ω else 0 |
| have hT (k : ℕ) : ∀ᵐ ω ∂P, Tendsto (fun n => T k n ω) atTop (𝓝 0) := by |
| apply ae_tendsto_zero_of_exponential_tail |
| intro ε hε |
| obtain ⟨C,a,hC,ha,N₀,hN₀⟩ := htail (k+1) ε (by positivity) hε |
| obtain ⟨D,hD,hDE⟩ := exponential_tail_extend_initial P |
| (fun n => {ω | ε ≤ |X n ω| ∧ Z n ω ≤ (k+1 : ℝ)}) hC ha N₀ hN₀ |
| refine ⟨D,a,hD,ha,fun n => ?_⟩ |
| have he : {ω | ε ≤ |T k n ω|}={ω | ε ≤ |X n ω| ∧ Z n ω ≤ (k+1 : ℝ)} := by |
| ext ω |
| by_cases hh : Z n ω ≤ (k+1 : ℝ) |
| · simp only [T,hh,ite_true,mem_ofPred_eq,and_true] |
| · simp only [T,hh,ite_false,abs_zero,mem_ofPred_eq,and_false,not_le_of_gt hε] |
| rw [he] |
| exact hDE n |
| filter_upwards [ae_all_iff.mpr hT,hbound] with ω hω hb |
| obtain ⟨K,hK⟩ := hb |
| obtain ⟨k,hk⟩ := exists_nat_gt K |
| have he : (fun n => T k n ω)=ᶠ[atTop] (fun n => X n ω) := by |
| filter_upwards [hK] with n hn |
| have hn' : Z n ω ≤ (k+1 : ℝ) := hn.trans (by exact le_trans hk.le (by linarith)) |
| simp only [T,ite_eq_left hn'] |
| exact (hω k).congr' he |
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| end InvariantIsing |
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| end |
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| end OAI |
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