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f0f05de | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import OAI.Combinatorics.BalancedRyser.Probability.BoundedDrift
import OAI.Combinatorics.BalancedRyser.Probability.FiniteHistory
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Preorder
open scoped MeasureTheory ProbabilityTheory BigOperators
namespace Balanced.FiniteHistory
variable {S : Type*} [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S]
theorem stronglyMeasurable_prefix (n : ℕ) (f : Prefix S n → ℝ) :
StronglyMeasurable[filtration n] (fun ω : Path S => f (frestrictLe n ω)) := by
apply Measurable.stronglyMeasurable
change Measurable[Filtration.piLE n] _
rw [Filtration.piLE_eq_comap_frestrictLe]
exact (measurable_of_countable f).comp (comap_measurable _)
theorem stronglyMeasurable_stepObservable (n : ℕ) (f : Prefix S n → S → ℝ) :
StronglyMeasurable[filtration (n + 1)] (stepObservable n f) := by
let g : Prefix S (n + 1) → ℝ := fun h =>
f (fun i => h ⟨i.1, Finset.mem_Iic.mpr (by have := Finset.mem_Iic.mp i.2; omega)⟩)
(h ⟨n + 1, Finset.mem_Iic.mpr le_rfl⟩)
exact stronglyMeasurable_prefix (n + 1) g
def additiveTrack (x₀ : ℝ) (f : (n : ℕ) → Prefix S n → S → ℝ)
(t : ℕ) (ω : Path S) : ℝ :=
x₀ + ∑ j ∈ Finset.range t, stepObservable j (f j) ω
omit [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] in
@[simp] theorem additiveTrack_zero (x₀ : ℝ)
(f : (n : ℕ) → Prefix S n → S → ℝ) (ω : Path S) :
additiveTrack x₀ f 0 ω = x₀ := by simp [additiveTrack]
omit [Fintype S] [MeasurableSpace S] [MeasurableSingletonClass S] in
@[simp] theorem additiveTrack_increment (x₀ : ℝ)
(f : (n : ℕ) → Prefix S n → S → ℝ) (t : ℕ) (ω : Path S) :
additiveTrack x₀ f (t + 1) ω - additiveTrack x₀ f t ω =
stepObservable t (f t) ω := by
simp only [additiveTrack, Finset.sum_range_succ]
ring
theorem stronglyMeasurable_additiveTrack (x₀ : ℝ)
(f : (n : ℕ) → Prefix S n → S → ℝ) (t : ℕ) :
StronglyMeasurable[filtration t] (additiveTrack x₀ f t) := by
apply StronglyMeasurable.add stronglyMeasurable_const
have hs := Finset.stronglyMeasurable_sum (Finset.range t) (fun j hj =>
(stronglyMeasurable_stepObservable j (f j)).mono
(filtration.mono (Nat.succ_le_of_lt (Finset.mem_range.mp hj))))
convert hs using 1
funext ω
simp only [Finset.sum_apply]
theorem condExp_additiveTrack_increment (step : ℕ → S → PMF S) (s₀ : S)
(x₀ : ℝ) (f : (n : ℕ) → Prefix S n → S → ℝ) (t : ℕ) :
(pathLaw step s₀)[fun ω => additiveTrack x₀ f (t + 1) ω -
additiveTrack x₀ f t ω | filtration t] =ᵐ[pathLaw step s₀]
fun ω => ∑ s, (step t (ω t) s).toReal * f t (frestrictLe t ω) s := by
simpa only [additiveTrack_increment] using condExp_stepObservable step s₀ t (f t)
theorem additiveTrack_drift_nonnegative (step : ℕ → S → PMF S) (s₀ : S)
(x₀ : ℝ) (f : (n : ℕ) → Prefix S n → S → ℝ) (T : ℕ)
(hmean : ∀ t < T, ∀ h : Prefix S t,
0 ≤ ∑ s, (step t (h ⟨t, Finset.mem_Iic.mpr le_rfl⟩) s).toReal * f t h s) :
∀ t ≤ T, ∀ᵐ ω ∂pathLaw step s₀,
0 ≤ ∑ j ∈ Finset.range t,
(pathLaw step s₀)[fun ω => additiveTrack x₀ f (j + 1) ω -
additiveTrack x₀ f j ω | filtration j] ω := by
intro t ht
have hall := ae_all_iff.2 (fun j => condExp_additiveTrack_increment step s₀ x₀ f j)
filter_upwards [hall] with ω hω
apply Finset.sum_nonneg
intro j hj
rw [hω j]
exact hmean j (lt_of_lt_of_le (Finset.mem_range.mp hj) ht) (frestrictLe j ω)
theorem additiveTrack_lower_tail (step : ℕ → S → PMF S) (s₀ : S)
(x₀ : ℝ) (f : (n : ℕ) → Prefix S n → S → ℝ) (T : ℕ) (L : ℝ)
(hT : 1 ≤ T) (hL : 0 < L)
(hbound : ∀ j < T, ∀ h s, |f j h s| ≤ L)
(hmean : ∀ j < T, ∀ h : Prefix S j,
0 ≤ ∑ s, (step j (h ⟨j, Finset.mem_Iic.mpr le_rfl⟩) s).toReal * f j h s)
(u : ℝ) (hu : 0 < u) (t : ℕ) (ht : t ≤ T) :
(pathLaw step s₀).real {ω | additiveTrack x₀ f t ω ≤ x₀ - u} ≤
2 * Real.exp (-u ^ 2 / (8 * (T : ℝ) * L ^ 2)) := by
have hc := boundedDriftConcentration_proof (pathLaw step s₀) filtration
(additiveTrack x₀ f) T L hT hL
(fun j _ => stronglyMeasurable_additiveTrack x₀ f j)
(fun j hj => Filter.Eventually.of_forall (fun ω => by
simpa only [additiveTrack_increment, stepObservable] using hbound j hj (frestrictLe j ω) (ω (j + 1))))
u hu t ht
refine le_trans ?_ hc
apply ENNReal.toReal_mono (measure_ne_top _ _)
apply measure_mono_ae
filter_upwards [additiveTrack_drift_nonnegative step s₀ x₀ f T hmean t ht]
with ω hω hbad
change additiveTrack x₀ f t ω ≤ x₀ - u at hbad
simp only [additiveTrack_zero]
have hneg : u ≤ -(additiveTrack x₀ f t ω - x₀ -
∑ j ∈ Finset.range t, (pathLaw step s₀)[fun ω =>
additiveTrack x₀ f (j + 1) ω - additiveTrack x₀ f j ω | filtration j] ω) := by
linarith
exact hneg.trans (neg_le_abs _)
end Balanced.FiniteHistory
end
end OAI
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