File size: 3,201 Bytes
f61e547 | 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 | import OAI.Probability.FreeIsing.Energy.TiltedVariance
import OAI.Probability.FreeIsing.Gaussian.LinearSignalTilt
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
lemma memLp_two_linearSignalTilt [IsProbabilityMeasure μ]
{q g : Ω → ℝ} {a : ℝ} (hq : Measurable q)
(hqb : ∀ᵐ x ∂μ, |q x| ≤ 1) (ha : |a| ≤ 1) (hg : MemLp g 2 μ) :
MemLp g 2 (linearSignalTilt μ q a) := by
have hgi := integrable_linearSignalTilt hq hqb ha (hg.integrable one_le_two)
exact (memLp_two_iff_integrable_sq hgi.aestronglyMeasurable).mpr
(integrable_linearSignalTilt hq hqb ha hg.integrable_sq)
lemma variance_linearSignalTilt_ge [IsProbabilityMeasure μ]
{q g : Ω → ℝ} {a : ℝ} (hq : Measurable q)
(hqb : ∀ᵐ x ∂μ, |q x| ≤ 1) (ha : |a| ≤ 1) (hg : MemLp g 2 μ)
(T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T μ μ)
(hqodd : ∀ᵐ x ∂μ, q (T x) = -q x)
(hgodd : ∀ᵐ x ∂μ, g (T x) = -g x) :
(1 - a ^ 2) * (∫ x, g x ^ 2 ∂μ) ≤ variance g (linearSignalTilt μ q a) := by
let := linearSignalTilt_probability hq hqb ha
(integral_eq_zero_of_ae_sign_reversal T hT hqodd)
apply variance_ge_of_binary_tilt_moments hg
(memLp_two_linearSignalTilt hq hqb ha hg) hq.aestronglyMeasurable hqb
· rw [integral_linearSignalTilt hq hqb ha (hg.integrable one_le_two),
integral_eq_zero_of_ae_sign_reversal T hT hgodd, zero_add]
congr 1
apply integral_congr_ae
filter_upwards [] with x
ring
· apply integral_linearSignalTilt_ae_even hq hqb ha hg.integrable_sq T hT hqodd
filter_upwards [hgodd] with x hx
rw [hx, neg_sq]
lemma variance_linearSignalTilt_ge_signal_moment [IsProbabilityMeasure μ]
{q g : Ω → ℝ} {a theta : ℝ} (hq : Measurable q)
(hqb : ∀ᵐ x ∂μ, |q x| ≤ 1) (ha : |a| ≤ 1) (hg : MemLp g 2 μ)
(T : Ω ≃ᵐ Ω) (hT : MeasurePreserving T μ μ)
(hqodd : ∀ᵐ x ∂μ, q (T x) = -q x)
(hgodd : ∀ᵐ x ∂μ, g (T x) = -g x)
(haTheta : a ^ 2 ≤ theta ^ 2) (hTheta : theta ^ 2 ≤ 1)
(hdom : ∀ᵐ x ∂μ, theta ^ 2 * q x ^ 2 ≤ g x ^ 2) :
(1 - theta ^ 2) * theta ^ 2 * (∫ x, q x ^ 2 ∂μ) ≤
variance g (linearSignalTilt μ q a) := by
have hqLp : MemLp q 2 μ :=
MemLp.of_bound hq.aestronglyMeasurable 1 (by simpa only [Real.norm_eq_abs] using hqb)
have henergy : theta ^ 2 * (∫ x, q x ^ 2 ∂μ) ≤ ∫ x, g x ^ 2 ∂μ := by
rw [← integral_const_mul]
exact integral_mono_ae (hqLp.integrable_sq.const_mul _) hg.integrable_sq hdom
have hnonneg : 0 ≤ ∫ x, g x ^ 2 ∂μ := integral_nonneg fun x => sq_nonneg _
calc
(1 - theta ^ 2) * theta ^ 2 * (∫ x, q x ^ 2 ∂μ) =
(1 - theta ^ 2) * (theta ^ 2 * (∫ x, q x ^ 2 ∂μ)) := by ring
_ ≤ (1 - theta ^ 2) * (∫ x, g x ^ 2 ∂μ) :=
mul_le_mul_of_nonneg_left henergy (sub_nonneg.mpr hTheta)
_ ≤ (1 - a ^ 2) * (∫ x, g x ^ 2 ∂μ) :=
mul_le_mul_of_nonneg_right (sub_le_sub_left haTheta 1) hnonneg
_ ≤ _ := variance_linearSignalTilt_ge hq hqb ha hg T hT hqodd hgodd
end Problem367
end
end OAI
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