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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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 | import OAI.Probability.InvariantIsing.Pressure.Concentration
namespace OAI
/-!
# Almost-sure concentration without independence across dimensions
A countable Borel–Cantelli argument upgrades Haar tails to almost-sure
concentration of the pressure about its medians.
-/
noncomputable section
open MeasureTheory Filter
open scoped Topology NNReal ENNReal
namespace InvariantIsing
/-- Summable exponential tails imply almost-sure convergence to zero. No
independence between the random variables is required. -/
lemma ae_tendsto_zero_of_exponential_tail {Ω : Type*} [MeasurableSpace Ω]
(μ : Measure Ω) [IsProbabilityMeasure μ] (f : ℕ → Ω → ℝ)
(htail : ∀ ε : ℝ, 0 < ε → ∃ C a : ℝ, 0 ≤ C ∧ 0 < a ∧
∀ n, μ.real {ω | ε ≤ |f n ω|} ≤ C * Real.exp (-a * (n : ℝ))) :
∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 0) := by
have hb (k : ℕ) : ∀ᵐ ω ∂μ, ∀ᶠ n in atTop,
¬ (1 / (k + 1 : ℝ) ≤ |f n ω|) := by
have hk : 0 < 1 / (k + 1 : ℝ) := by positivity
obtain ⟨C, a, hC, ha, ht⟩ := htail _ hk
have hsum : Summable (fun n : ℕ => C * Real.exp (-a * (n : ℝ))) := by
simp_rw [mul_comm (-a), Real.exp_nat_mul]
exact (summable_geometric_of_lt_one (Real.exp_pos (-a)).le
(Real.exp_lt_one_iff.mpr (neg_neg_of_pos ha))).mul_left C
let q : ℕ → ℝ≥0 := fun n => NNReal.mk (C * Real.exp (-a * (n : ℝ)))
(mul_nonneg hC (Real.exp_pos _).le)
have hq : Summable q := (NNReal.summable_coe (f := q)).mp hsum
have hfinite : (∑' n, (q n : ENNReal)) ≠ ∞ :=
ENNReal.tsum_coe_ne_top_iff_summable.mpr hq
have hbound (n : ℕ) : μ {ω | 1 / (k + 1 : ℝ) ≤ |f n ω|} ≤ (q n : ENNReal) := by
rw [← ENNReal.ofReal_toReal (measure_ne_top μ {ω | 1 / (k + 1 : ℝ) ≤ |f n ω|})]
change ENNReal.ofReal (μ.real {ω | 1 / (k + 1 : ℝ) ≤ |f n ω|}) ≤ _
simpa only [q, ENNReal.ofReal_eq_coe_nnreal (mul_nonneg hC (Real.exp_pos _).le)] using
ENNReal.ofReal_le_ofReal (ht n)
exact ae_eventually_notMem (s := fun n => {ω | 1 / (k + 1 : ℝ) ≤ |f n ω|})
(ne_top_of_le_ne_top hfinite (ENNReal.tsum_le_tsum hbound))
filter_upwards [ae_all_iff.mpr hb] with ω hω
apply Metric.tendsto_atTop.mpr
intro ε hε
obtain ⟨k, hk⟩ := exists_nat_one_div_lt hε
obtain ⟨N, hN⟩ := eventually_atTop.mp (hω k)
refine ⟨N, ?_⟩
intro n hn
simpa only [Real.dist_eq, sub_zero] using lt_trans (lt_of_not_ge (hN n hn)) hk
/-- For an arbitrary joint realization of the Haar matrices, pressure is almost
surely close to a deterministic sequence of medians. -/
theorem ae_pressure_sub_median_tendsto_zero (hpub : HaarConcentrationInput)
{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
(U : (n : ℕ) → Ω → SpecialOrthogonal (n + 3))
(hU : ∀ n, Measurable (U n))
(hHaar : ∀ n, (P.map (U n)).IsMulLeftInvariant)
(eig c : (n : ℕ) → Fin (n + 3) → ℝ)
(K : ℝ) (hK : 0 < K) (heig : ∀ n i, |eig n i| ≤ K) :
∃ m : ℕ → ℝ, ∀ᵐ ω ∂P,
Tendsto (fun n => rotatedPressure (eig n) (specialRotation (U n ω)) (c n) - m n)
atTop (𝓝 0) := by
obtain ⟨C, a, hC, ha, ht⟩ := haar_pressure_median_tail hpub
have hm (n : ℕ) : ∃ m : ℝ,
∀ r : ℝ, 0 < r →
(P.map (U n)).real {V | r ≤ |rotatedPressure (eig n) (specialRotation V) (c n) - m|} ≤
C * Real.exp (-a * (n + 3 : ℝ) * r ^ 2 / K ^ 2) := by
have hp : IsProbabilityMeasure (P.map (U n)) :=
(Measure.isProbabilityMeasure_map_iff (hU n).aemeasurable).mpr inferInstance
obtain ⟨m, _, _, hm⟩ := ht (n + 3) (by omega) (P.map (U n)) hp (hHaar n)
(eig n) (c n) K hK (heig n)
exact ⟨m, by simpa only [Nat.cast_add, Nat.cast_ofNat] using hm⟩
choose m hm using hm
refine ⟨m, ae_tendsto_zero_of_exponential_tail P _ ?_⟩
intro ε hε
refine ⟨C, a * ε ^ 2 / K ^ 2, hC.le, by positivity, ?_⟩
intro n
have hs : MeasurableSet {V : SpecialOrthogonal (n + 3) |
ε ≤ |rotatedPressure (eig n) (specialRotation V) (c n) - m n|} :=
measurableSet_le measurable_const ((measurable_rotatedPressure _ _).sub_const _).abs
have hb := hm n ε hε
rw [measureReal_def, Measure.map_apply (hU n) hs] at hb
change P.real {ω | ε ≤ |rotatedPressure (eig n) (specialRotation (U n ω)) (c n) - m n|} ≤ _ at hb
refine hb.trans ?_
apply mul_le_mul_of_nonneg_left _ hC.le
apply Real.exp_le_exp.mpr
have hnonneg : 0 ≤ a * ε ^ 2 / K ^ 2 := by positivity
calc
-a * ((n : ℝ) + 3) * ε ^ 2 / K ^ 2 =
-(a * ε ^ 2 / K ^ 2) * ((n : ℝ) + 3) := by ring
_ ≤ -(a * ε ^ 2 / K ^ 2) * (n : ℝ) :=
mul_le_mul_of_nonpos_left (by linarith) (neg_nonpos.mpr hnonneg)
end InvariantIsing
end
end OAI
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