ProbabilityTheory.hasSubexponentialMGF_iff_kernel
Lemma
No docstring.
Types
-
Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
Given
-
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 -
b : ℝ
Then
HasSubexponentialMGF X V b μ ↔ Kernel.HasSubexponentialMGF X V b (Kernel.const Unit μ) (MeasureTheory.Measure.dirac ())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.
Iff : Prop → Prop → PropIf and only if, or logical bi-implication. `a ↔ b` means that `a` implies `b` and vice versa. By `propext`, this implies that `a` and `b` are equal and hence any expression involving `a` is equivalent to the corresponding expression with `b` instead. Conventions for notations in identifiers: * The recommended spelling of `↔` in identifiers is `iff`. * The recommended spelling of `<->` in identifiers is `iff` (prefer `↔` over `<->`).
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
ProbabilityTheory.Kernel.HasSubexponentialMGF : {Ω : Type u_1} →
{Ω' : Type u_2} →
{mΩ : MeasurableSpace Ω} →
{mΩ' : MeasurableSpace Ω'} →
(Ω → ℝ) →
ℝ →
ℝ →
ProbabilityTheory.Kernel Ω' Ω →
autoParam (MeasureTheory.Measure Ω') ProbabilityTheory.Kernel.HasSubexponentialMGF._auto_1 → PropA random variable `X` has a sub-exponential moment-generating function with parameters `(V, b)` with respect to a kernel `κ` and a measure `ν` if for every `t` with `b * |t| ≤ 1`, `exp (t * X)` is integrable with respect to `κ ∘ₘ ν` and, for `ν`-almost all `ω'`, the moment-generating function of `X` with respect to `κ ω'` is bounded by `exp (V * t ^ 2 / 2)`. For `b = 0` this is `Kernel.HasSubgaussianMGF X V κ ν`.Go to its page
ProbabilityTheory.Kernel.const : (α : Type u_4) →
{β : Type u_5} →
[inst : MeasurableSpace α] → {x : MeasurableSpace β} → MeasureTheory.Measure β → ProbabilityTheory.Kernel α βConstant kernel, which always returns the same measure.
Unit : TypeThe canonical type with one element. This element is written `()`. `Unit` has a number of uses: * It can be used to model control flow that returns from a function call without providing other information. * Monadic actions that return `Unit` have side effects without computing values. * In polymorphic types, it can be used to indicate that no data is to be stored in a particular field.
MeasureTheory.Measure.dirac : {α : Type u_1} → [inst : MeasurableSpace α] → α → MeasureTheory.Measure αThe dirac measure.
Unit.unit : UnitThe only element of the unit type. It can be written as an empty tuple: `()`.
Code
lemma hasSubexponentialMGF_iff_kernel :
HasSubexponentialMGF X V b μ
↔ Kernel.HasSubexponentialMGF X V b (Kernel.const Unit μ) (Measure.dirac ())Proof
⟨fun ⟨h1, h2⟩ ↦ ⟨by simpa, by simpa⟩, fun ⟨h1, h2⟩ ↦ ⟨by simpa using h1, by simpa using h2⟩⟩
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: 2 project declarations, 66 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.