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namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
theorem norm_integral_weighted_exp_remainder_le
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω)
(Z : Ω → ℂ) (D : Ω → ℝ)
(hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1)
(hD3 : Integrable (fun ω => |D ω| ^ 3) μ) :
‖∫ ω, Z ω * (Complex.exp ((D ω : ℂ) * Complex.I) - 1 -
(D ω : ℂ) * Complex.I + (D ω : ℂ) ^ 2 / 2) ∂μ‖ ≤
4 * ∫ ω, |D ω| ^ 3 ∂μ := by
rw [← integral_const_mul]
apply norm_integral_le_of_norm_le (hD3.const_mul 4)
filter_upwards [hZ] with ω hω
rw [norm_mul]
calc
_ ≤ 1 * ‖Complex.exp ((D ω : ℂ) * Complex.I) - 1 -
(D ω : ℂ) * Complex.I + (D ω : ℂ) ^ 2 / 2‖ :=
mul_le_mul_of_nonneg_right hω (norm_nonneg _)
_ ≤ 4 * |D ω| ^ 3 := by
simpa only [one_mul] using ExponentialTaylor.norm_exp_I_taylor_two_le (D ω)
theorem norm_integral_weighted_characteristic_step
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsFiniteMeasure μ]
(Z : Ω → ℂ) (D : Ω → ℝ) (h : ℝ)
(hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1)
(hDm : Measurable D) (hD : Integrable D μ)
(hD2 : Integrable (fun ω => D ω ^ 2) μ)
(hD3 : Integrable (fun ω => |D ω| ^ 3) μ)
(hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) :
‖(∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) -
(1 - (h : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤
4 * ∫ ω, |D ω| ^ 3 ∂μ +
(1 / 2 : ℝ) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ‖ := by
let R : Ω → ℂ := fun ω => Complex.exp ((D ω : ℂ) * Complex.I) - 1 -
(D ω : ℂ) * Complex.I + (D ω : ℂ) ^ 2 / 2
have hZint : Integrable Z μ :=
(integrable_const (1 : ℝ)).mono' hZm hZ
have hRm : Measurable R := by dsimp [R]; fun_prop
have hRint : Integrable R μ := (hD3.const_mul 4).mono' hRm.aestronglyMeasurable
(Filter.Eventually.of_forall fun ω => ExponentialTaylor.norm_exp_I_taylor_two_le (D ω))
have hZR : Integrable (fun ω => Z ω * R ω) μ := hRint.bdd_mul hZm hZ
have hZD : Integrable (fun ω => Z ω * (D ω : ℂ)) μ :=
hD.ofReal.bdd_mul hZm hZ
have hZQ : Integrable (fun ω => Z ω * ((D ω : ℂ) ^ 2 - h)) μ := by
have hq : Integrable (fun ω => (D ω : ℂ) ^ 2 - h) μ := by
have hd2 : Integrable (fun ω => (D ω : ℂ) ^ 2) μ := by
convert hD2.ofReal using 1
funext ω
exact (Complex.ofReal_pow (D ω) 2).symm
exact hd2.sub (integrable_const (h : ℂ))
exact hq.bdd_mul hZm hZ
have hE : Integrable (fun ω => Z ω * Complex.exp ((D ω : ℂ) * Complex.I)) μ := by
apply hZint.mul_bdd (c := 1)
· apply Measurable.aestronglyMeasurable
fun_prop
· exact Filter.Eventually.of_forall (fun ω => by simp)
have halg : (fun ω => Z ω * Complex.exp ((D ω : ℂ) * Complex.I) -
(1 - (h : ℂ) / 2) * Z ω) =
(fun ω => Z ω * R ω + (Z ω * (D ω : ℂ)) * Complex.I -
(Z ω * ((D ω : ℂ) ^ 2 - h)) * (1 / 2 : ℂ)) := by
funext ω
dsimp [R]
ring
have hid : (∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) -
(1 - (h : ℂ) / 2) * ∫ ω, Z ω ∂μ =
(∫ ω, Z ω * R ω ∂μ) -
(∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ) * (1 / 2 : ℂ) := by
rw [← integral_const_mul, ← integral_sub hE (hZint.const_mul _), halg]
have hsum : Integrable (fun ω => Z ω * R ω + Z ω * (D ω : ℂ) * Complex.I) μ :=
hZR.add (hZD.mul_const Complex.I)
rw [integral_sub hsum (hZQ.mul_const (1 / 2 : ℂ)),
integral_add hZR (hZD.mul_const Complex.I), integral_mul_const, integral_mul_const, hzero]
simp
rw [hid]
calc
_ ≤ ‖∫ ω, Z ω * R ω ∂μ‖ +
‖(∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ) * (1 / 2 : ℂ)‖ := norm_sub_le _ _
_ ≤ (4 * ∫ ω, |D ω| ^ 3 ∂μ) +
(1 / 2 : ℝ) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ‖ := by
have hr := norm_integral_weighted_exp_remainder_le μ Z D hZ hD3
have hn : ‖(1 / 2 : ℂ)‖ = (1 / 2 : ℝ) := by norm_num
rw [norm_mul, hn, mul_comm _ (1 / 2 : ℝ)]
exact add_le_add hr le_rfl
theorem norm_gaussian_coefficient_sub_euler_le {h : ℝ}
(h0 : 0 ≤ h) (h2 : h ≤ 2) :
‖Complex.exp (-(h : ℂ) / 2) - (1 - (h : ℂ) / 2)‖ ≤ h ^ 2 / 4 := by
have hn : ‖-(h : ℂ) / 2‖ = h / 2 := by
simp [Real.norm_eq_abs, abs_of_nonneg h0]
have hb := Complex.norm_exp_sub_one_sub_id_le
(x := -(h : ℂ) / 2) (by rw [hn]; linarith)
rw [hn] at hb
convert hb using 1 <;> congr 1 <;> ring
theorem norm_gaussian_coefficient_sub_euler_le_global {h : ℝ} (h0 : 0 ≤ h) :
‖Complex.exp (-(h : ℂ) / 2) - (1 - (h : ℂ) / 2)‖ ≤ h ^ 2 / 4 := by
have hb := ExponentialTaylor.norm_exp_neg_sub_one_add_le_sq
(show 0 ≤ h / 2 by positivity)
convert hb using 1 <;> push_cast <;> congr 1 <;> ring_nf
theorem norm_integral_weighted_characteristic_gaussian_step_all
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(Z : Ω → ℂ) (D : Ω → ℝ) {h : ℝ} (h0 : 0 ≤ h)
(hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1)
(hDm : Measurable D) (hD : Integrable D μ)
(hD2 : Integrable (fun ω => D ω ^ 2) μ)
(hD3 : Integrable (fun ω => |D ω| ^ 3) μ)
(hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) :
‖(∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) -
Complex.exp (-(h : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤
4 * ∫ ω, |D ω| ^ 3 ∂μ +
(1 / 2 : ℝ) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ‖ + h ^ 2 / 4 := by
have he := norm_integral_weighted_characteristic_step μ Z D h
hZm hZ hDm hD hD2 hD3 hzero
have hZi : ‖∫ ω, Z ω ∂μ‖ ≤ 1 := by
simpa using norm_integral_le_of_norm_le (integrable_const (1 : ℝ)) hZ
have hc : ‖(1 - (h : ℂ) / 2) - Complex.exp (-(h : ℂ) / 2)‖ ≤ h ^ 2 / 4 := by
rw [norm_sub_rev]
exact norm_gaussian_coefficient_sub_euler_le_global h0
have halg : (∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) -
Complex.exp (-(h : ℂ) / 2) * ∫ ω, Z ω ∂μ =
((∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) -
(1 - (h : ℂ) / 2) * ∫ ω, Z ω ∂μ) +
((1 - (h : ℂ) / 2) - Complex.exp (-(h : ℂ) / 2)) * ∫ ω, Z ω ∂μ := by ring
rw [halg]
apply (norm_add_le _ _).trans
apply add_le_add he
rw [norm_mul]
calc
_ ≤ ‖(1 - (h : ℂ) / 2) - Complex.exp (-(h : ℂ) / 2)‖ * 1 :=
mul_le_mul_of_nonneg_left hZi (norm_nonneg _)
_ ≤ h ^ 2 / 4 := by simpa only [mul_one] using hc
theorem norm_integral_weighted_characteristic_gaussian_step
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(Z : Ω → ℂ) (D : Ω → ℝ) {h : ℝ} (h0 : 0 ≤ h) (_h2 : h ≤ 2)
(hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1)
(hDm : Measurable D) (hD : Integrable D μ)
(hD2 : Integrable (fun ω => D ω ^ 2) μ)
(hD3 : Integrable (fun ω => |D ω| ^ 3) μ)
(hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) :
‖(∫ ω, Z ω * Complex.exp ((D ω : ℂ) * Complex.I) ∂μ) -
Complex.exp (-(h : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤
4 * ∫ ω, |D ω| ^ 3 ∂μ +
(1 / 2 : ℝ) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - h) ∂μ‖ + h ^ 2 / 4 :=
norm_integral_weighted_characteristic_gaussian_step_all μ Z D h0
hZm hZ hDm hD hD2 hD3 hzero
theorem norm_integral_weighted_characteristic_gaussian_step_all_mul
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(Z : Ω → ℂ) (D : Ω → ℝ) (u : ℝ) {v : ℝ}
(hv : 0 ≤ v)
(hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1)
(hDm : Measurable D) (hD : Integrable D μ)
(hD2 : Integrable (fun ω => D ω ^ 2) μ)
(hD3 : Integrable (fun ω => |D ω| ^ 3) μ)
(hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) :
‖(∫ ω, Z ω * Complex.exp (((u * D ω : ℝ) : ℂ) * Complex.I) ∂μ) -
Complex.exp (-((u ^ 2 * v : ℝ) : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤
4 * |u| ^ 3 * (∫ ω, |D ω| ^ 3 ∂μ) +
(u ^ 2 / 2) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - v) ∂μ‖ +
u ^ 4 * v ^ 2 / 4 := by
have hs2 : Integrable (fun ω => (u * D ω) ^ 2) μ := by
simpa only [mul_pow] using hD2.const_mul (u ^ 2)
have hs3 : Integrable (fun ω => |u * D ω| ^ 3) μ := by
simpa only [abs_mul, mul_pow] using hD3.const_mul (|u| ^ 3)
have hsz : ∫ ω, Z ω * ((u * D ω : ℝ) : ℂ) ∂μ = 0 := by
calc
_ = (u : ℂ) * ∫ ω, Z ω * (D ω : ℂ) ∂μ := by
rw [← integral_const_mul]
congr 1
funext ω
push_cast
ring
_ = 0 := by rw [hzero, mul_zero]
have hs := norm_integral_weighted_characteristic_gaussian_step_all μ Z
(fun ω => u * D ω) (mul_nonneg (sq_nonneg u) hv)
hZm hZ (hDm.const_mul u) (hD.const_mul u) hs2 hs3 hsz
have hthird : (∫ ω, |u * D ω| ^ 3 ∂μ) =
|u| ^ 3 * ∫ ω, |D ω| ^ 3 ∂μ := by
simp only [abs_mul, mul_pow, integral_const_mul]
have hquad : (∫ ω, Z ω * (((u * D ω : ℝ) : ℂ) ^ 2 - (u ^ 2 * v : ℝ)) ∂μ) =
(u : ℂ) ^ 2 * ∫ ω, Z ω * ((D ω : ℂ) ^ 2 - v) ∂μ := by
rw [← integral_const_mul]
congr 1
funext ω
push_cast
ring
rw [hthird, hquad, norm_mul] at hs
have hn : ‖(u : ℂ) ^ 2‖ = u ^ 2 := by simp [norm_pow, sq_abs]
rw [hn] at hs
convert hs using 1
ring
theorem norm_integral_weighted_characteristic_gaussian_step_mul
{Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) [IsProbabilityMeasure μ]
(Z : Ω → ℂ) (D : Ω → ℝ) (u : ℝ) {v : ℝ}
(hv : 0 ≤ v) (_huv : u ^ 2 * v ≤ 2)
(hZm : AEStronglyMeasurable Z μ) (hZ : ∀ᵐ ω ∂μ, ‖Z ω‖ ≤ 1)
(hDm : Measurable D) (hD : Integrable D μ)
(hD2 : Integrable (fun ω => D ω ^ 2) μ)
(hD3 : Integrable (fun ω => |D ω| ^ 3) μ)
(hzero : ∫ ω, Z ω * (D ω : ℂ) ∂μ = 0) :
‖(∫ ω, Z ω * Complex.exp (((u * D ω : ℝ) : ℂ) * Complex.I) ∂μ) -
Complex.exp (-((u ^ 2 * v : ℝ) : ℂ) / 2) * ∫ ω, Z ω ∂μ‖ ≤
4 * |u| ^ 3 * (∫ ω, |D ω| ^ 3 ∂μ) +
(u ^ 2 / 2) * ‖∫ ω, Z ω * ((D ω : ℂ) ^ 2 - v) ∂μ‖ +
u ^ 4 * v ^ 2 / 4 :=
norm_integral_weighted_characteristic_gaussian_step_all_mul μ Z D u hv
hZm hZ hDm hD hD2 hD3 hzero
end Problem367
end
end OAI
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