Theorems · Theorem · probability
ProbabilityTheory.mgf_le_of_mem_Icc_of_integral_eq_zero
∀ {Ω : Type u_1} {mΩ : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {X : Ω → ℝ}
[MeasureTheory.IsProbabilityMeasure μ] {a b t : ℝ},
AEMeasurable X μ →
(∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) →
∫ (x : Ω), X x ∂μ = 0 → 0 < t → ProbabilityTheory.mgf X μ t ≤ Real.exp ((↑‖b - a‖₊ / 2) ^ 2 * t ^ 2 / 2)- Defined in
- Mathlib.Probability.Moments.SubGaussian
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 297 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites44
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- MeasurableSpacestatement and proof · cited by 13,106
- MeasureTheory.Measurestatement and proof · cited by 10,939
- Set.ofPredproof · cited by 6,101
- Filter.Eventuallystatement and proof · cited by 3,134
- Compl.complproof · cited by 2,925
- one_mulproof · cited by 2,841
- MeasureTheory.aestatement and proof · cited by 2,352
- Nat.cast_zeroproof · cited by 1,870
- absproof · cited by 1,814
- MeasureTheory.integralstatement and proof · cited by 1,779
Cited by1
Results whose statement or proof uses this declaration.
- ProbabilityTheory.hasSubgaussianMGF_of_mem_Icc_of_integral_eq_zeroproof · cited by 1