File size: 3,088 Bytes
a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Probability.InvariantIsing.Pressure.AlmostSure
namespace OAI
/-! 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
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))
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
end InvariantIsing
end
end OAI
|