`ProbabilityTheory.HasSubexponentialMGF`🔗

Structure

From the authors

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.

Types

Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.

Given

X : Ω → ℝ
V : ℝ
b : ℝ
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.

Result

Prop

Fields

integrable_exp_mul : ∀ (t : ℝ), b * |t| ≤ 1 → MeasureTheory.Integrable (fun ω => Real.exp (t * X ω)) μIntegrable f μ means that f is measurable and that the integral ∫⁻ a, ‖f a‖ ∂μ is finite.
mgf_le : ∀ (t : ℝ), b * |t| ≤ 1 → mgf X μ t ≤ Real.exp (V * t ^ 2 / 2)

MeasurableSpace : Type u_6 → Type u_6

A measurable space is a space equipped with a σ-algebra.

Real : Type

The type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational
numbers.

MeasureTheory.Measure : (α : Type u_5) → [MeasurableSpace α] → Type u_5

A 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`.

MeasureTheory.Integrable : {ε : Type u_5} →
  [inst : TopologicalSpace ε] →
    [ContinuousENorm ε] →
      {α : Type u_7} →
        {x : MeasurableSpace α} → (α → ε) → autoParam (MeasureTheory.Measure α) MeasureTheory.Integrable._auto_1 → Prop

`Integrable f μ` means that `f` is measurable and that the integral `∫⁻ a, ‖f a‖ ∂μ` is finite.
`Integrable f` means `Integrable f volume`.

HMul.hMul : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HMul α β γ] → α → β → γ

`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `*` in identifiers is `mul`.

abs : {α : Type u_1} → [Lattice α] → [AddGroup α] → α → α

`abs a`, denoted `|a|`, is the absolute value of `a`

LE.le : {α : Type u} → [self : LE α] → α → α → Prop

The less-equal relation: `x ≤ y` 

Conventions for notations in identifiers:

 * The recommended spelling of `≤` in identifiers is `le`.

Real.exp : ℝ → ℝ

The real exponential function, defined as the real part of the complex exponential

ProbabilityTheory.mgf : {Ω : Type u_1} → {m : MeasurableSpace Ω} → (Ω → ℝ) → MeasureTheory.Measure Ω → ℝ → ℝ

Moment-generating function of a real random variable `X`: `fun t => μ[exp(t*X)]`.

HPow.hPow : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HPow α β γ] → α → β → γ

`a ^ b` computes `a` to the power of `b`.
The meaning of this notation is type-dependent. 

Conventions for notations in identifiers:

 * The recommended spelling of `^` in identifiers is `pow`.

HDiv.hDiv : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HDiv α β γ] → α → β → γ

`a / b` computes the result of dividing `a` by `b`.
The meaning of this notation is type-dependent.
* For most types like `Nat`, `Int`, `Rat`, `Real`, `a / 0` is defined to be `0`.
* For `Nat`, `a / b` rounds downwards.
* For `Int`, `a / b` rounds downwards if `b` is positive or upwards if `b` is negative.
  It is implemented as `Int.ediv`, the unique function satisfying
  `a % b + b * (a / b) = a` and `0 ≤ a % b < natAbs b` for `b ≠ 0`.
  Other rounding conventions are available using the functions
  `Int.fdiv` (floor rounding) and `Int.tdiv` (truncation rounding).
* For `Float`, `a / 0` follows the IEEE 754 semantics for division,
  usually resulting in `inf` or `nan`. 

Conventions for notations in identifiers:

 * The recommended spelling of `/` in identifiers is `div`.

Code

structure HasSubexponentialMGF (X : Ω → ℝ) (V b : ℝ) (μ : Measure Ω := by volume_tac) :
    Prop where
  integrable_exp_mul : ∀ t : ℝ, b * |t| ≤ 1 → Integrable (fun ω ↦ exp (t * X ω)) μ
  mgf_le : ∀ t : ℝ, b * |t| ≤ 1 → mgf X μ t ≤ exp (V * t ^ 2 / 2)

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

Nothing to draw. Its statement rests on no other declaration in this project, and names nothing from a package left unaudited — so the graph is this declaration alone. That is the answer, not a missing picture.

Audit surface: 0 project declarations, 52 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.

ProbabilityTheory.HasSubexponentialMGF🔗

`ProbabilityTheory.HasSubexponentialMGF`🔗