ProbabilityTheory.HasSubexponentialMGF.hasSubgaussianMGF
Lemma
No docstring.
Types
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Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
Given
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μ : MeasureTheory.Measure ΩA measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. -
X : Ω → ℝ -
V : ℝ
Assuming
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h : HasSubexponentialMGF X V 0 μXhas a sub-exponential moment generating function with parameters(V, b): for everytwithb * |t| ≤ 1,exp (t * X)is integrable andmgf X μ t ≤ exp (V * t ^ 2 / 2). -
hV : 0 ≤ V
Then
HasSubgaussianMGF X ⟨V, hV⟩ μX has a sub-Gaussian moment-generating function with parameter c with respect to a measure μ if for all t : ℝ, exp (t * X) is μ-integrable and the moment-generating funct…MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5A measure is defined to be an outer measure that is countably additive on measurable sets, with the additional assumption that the outer measure is the canonical extension of the restricted measure. The measure of a set `s`, denoted `μ s`, is an extended nonnegative real. The real-valued version is written `μ.real s`.
Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
ProbabilityTheory.HasSubexponentialMGF : {Ω : Type u_1} →
{mΩ : MeasurableSpace Ω} →
(Ω → ℝ) → ℝ → ℝ → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasSubexponentialMGF._auto_1 → Prop`X` has a sub-exponential moment generating function with parameters `(V, b)`: for every `t` with `b * |t| ≤ 1`, `exp (t * X)` is integrable and `mgf X μ t ≤ exp (V * t ^ 2 / 2)`. For `b = 0` this is `HasSubgaussianMGF X V μ`. This is equivalent to `Kernel.HasSubexponentialMGF X V b (Kernel.const Unit μ) (Measure.dirac ())`, as proved in `hasSubexponentialMGF_iff_kernel`.Go to its page
LE.le : {α : Type u} → [self : LE α] → α → α → PropThe less-equal relation: `x ≤ y` Conventions for notations in identifiers: * The recommended spelling of `≤` in identifiers is `le`.
ProbabilityTheory.HasSubgaussianMGF : {Ω : Type u_1} →
{mΩ : MeasurableSpace Ω} →
(Ω → ℝ) → NNReal → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasSubgaussianMGF._auto_1 → PropA random variable `X` has a sub-Gaussian moment-generating function with parameter `c` with respect to a measure `μ` if for all `t : ℝ`, `exp (t * X)` is `μ`-integrable and the moment-generating function of `X` is bounded by `exp (c * t ^ 2 / 2)` for all `t : ℝ`. This implies in particular that `X` has expectation 0. This is equivalent to `Kernel.HasSubgaussianMGF X c (Kernel.const Unit μ) (Measure.dirac ())`, as proved in `HasSubgaussianMGF_iff_kernel`. Properties about sub-Gaussian moment-generating functions should be proved first for `Kernel.HasSubgaussianMGF` when possible.
Subtype.mk : {α : Sort u} → {p : α → Prop} → (val : α) → p val → Subtype pCode
lemma hasSubgaussianMGF (h : HasSubexponentialMGF X V 0 μ) (hV : 0 ≤ V) :
HasSubgaussianMGF X ⟨V, hV⟩ μ where
integrable_exp_mul tProof
h.integrable_exp_mul t (by simp) mgf_le t := h.mgf_le t (by simp)
Meaning last changed in v4.34.0-rc2-76-g565f652 (2026-09-10).
Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.
Dependency graph
Audit surface: 1 project declarations, 56 external constants
✓ Proved: no sorry anywhere in its closure
This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.