| import Mathlib.MeasureTheory.Function.ConditionalExpectation.PullOut |
| import Mathlib.Probability.Moments.SubGaussian |
|
|
| namespace OAI |
|
|
| open MeasureTheory ProbabilityTheory Real |
| open scoped ENNReal NNReal |
|
|
| namespace Balanced |
| open scoped _root_.Balanced |
|
|
| theorem integral_mul_le_of_condExp_le |
| {Ω : Type*} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} |
| [IsProbabilityMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ) |
| {F G : Ω → ℝ} {K : ℝ} |
| (hFm : StronglyMeasurable[m] F) (hF : Integrable F μ) |
| (hG : Integrable G μ) (hFG : Integrable (F * G) μ) |
| (hFnonneg : ∀ᵐ ω ∂μ, 0 ≤ F ω) |
| (hGcond : ∀ᵐ ω ∂μ, μ[G | m] ω ≤ K) : |
| (∫ ω, F ω * G ω ∂μ) ≤ K * ∫ ω, F ω ∂μ := by |
| have hpull := condExp_mul_of_stronglyMeasurable_left hFm hFG hG |
| have hprod : Integrable (fun ω => F ω * μ[G | m] ω) μ := |
| integrable_condExp.congr hpull |
| calc |
| (∫ ω, F ω * G ω ∂μ) = ∫ ω, μ[F * G | m] ω ∂μ := |
| (integral_condExp hm).symm |
| _ = ∫ ω, F ω * μ[G | m] ω ∂μ := integral_congr_ae hpull |
| _ ≤ ∫ ω, F ω * K ∂μ := by |
| apply integral_mono_ae hprod (hF.mul_const K) |
| filter_upwards [hFnonneg, hGcond] with ω hFω hGω |
| exact mul_le_mul_of_nonneg_left hGω hFω |
| _ = K * ∫ ω, F ω ∂μ := by rw [integral_mul_const]; ring |
|
|
| theorem mgf_add_le_of_condExp_exp_le |
| {Ω : Type*} {mΩ : MeasurableSpace Ω} {μ : Measure Ω} |
| [IsProbabilityMeasure μ] {m : MeasurableSpace Ω} (hm : m ≤ mΩ) |
| {S Z : Ω → ℝ} {θ K : ℝ} |
| (hSm : StronglyMeasurable[m] S) |
| (hSexp : Integrable (fun ω => exp (θ * S ω)) μ) |
| (hZexp : Integrable (fun ω => exp (θ * Z ω)) μ) |
| (hSumexp : Integrable (fun ω => exp (θ * (S ω + Z ω))) μ) |
| (hZcond : ∀ᵐ ω ∂μ, μ[fun ω => exp (θ * Z ω) | m] ω ≤ K) : |
| mgf (S + Z) μ θ ≤ K * mgf S μ θ := by |
| have hproduct : Integrable ((fun ω => exp (θ * S ω)) * |
| (fun ω => exp (θ * Z ω))) μ := by |
| convert hSumexp using 1 |
| ext ω |
| simp only [Pi.mul_apply, mul_add, exp_add] |
| simpa only [mgf, Pi.add_apply, mul_add, exp_add] using |
| (integral_mul_le_of_condExp_le hm |
| (Real.continuous_exp.comp_stronglyMeasurable (hSm.const_mul θ)) |
| hSexp hZexp hproduct |
| (Filter.Eventually.of_forall fun ω => (exp_pos (θ * S ω)).le) hZcond) |
|
|
| end Balanced |
|
|
| end OAI |
|
|
