ProbabilityTheory.Kernel.HasSubexponentialMGF
From the authors
A 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.
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Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra. -
Ω' : Type u_2mΩ' : MeasurableSpace Ω'
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X : Ω → ℝ -
V : ℝ -
b : ℝ -
κ : Kernel Ω' ΩA kernel from a measurable spaceαto another measurable spaceβis a measurable functionκ : α → Measure β. -
ν : MeasureTheory.Measure Ω'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.
Prop-
integrable_exp_mul : ∀ (t : ℝ), b * |t| ≤ 1 → MeasureTheory.Integrable (fun ω => Real.exp (t * X ω)) (ν.bind ⇑κ)Integrable f μmeans thatfis measurable and that the integral∫⁻ a, ‖f a‖ ∂μis finite. -
mgf_le : ∀ᵐ (ω' : Ω') ∂ν, ∀ (t : ℝ), b * |t| ≤ 1 → mgf X (κ ω') t ≤ Real.exp (V * t ^ 2 / 2)f.Eventually por∀ᶠ x in f, p xmean that{x | p x} ∈ f.
MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
Real : TypeThe type `ℝ` of real numbers constructed as equivalence classes of Cauchy sequences of rational numbers.
ProbabilityTheory.Kernel : (α : Type u_1) → (β : Type u_2) → [MeasurableSpace α] → [MeasurableSpace β] → Type (max u_1 u_2)A kernel from a measurable space `α` to another measurable space `β` is a measurable function `κ : α → Measure β`. The measurable space structure on `MeasureTheory.Measure β` is given by `MeasureTheory.Measure.instMeasurableSpace`. A map `κ : α → MeasureTheory.Measure β` is measurable iff `∀ s : Set β, MeasurableSet s → Measurable (fun a ↦ κ a s)`.
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`.
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 → PropThe less-equal relation: `x ≤ y` Conventions for notations in identifiers: * The recommended spelling of `≤` in identifiers is `le`.
Real.exp : ℝ → ℝThe real exponential function, defined as the real part of the complex exponential
MeasureTheory.Measure.bind : {α : Type u_1} →
{β : Type u_2} →
{mα : MeasurableSpace α} →
{mβ : MeasurableSpace β} → MeasureTheory.Measure α → (α → MeasureTheory.Measure β) → MeasureTheory.Measure βMonadic bind on `Measure`, only works in the category of measurable spaces and measurable functions. When the function `f` is not measurable the result is not well defined.
Filter.Eventually : {α : Type u_1} → (α → Prop) → Filter α → Prop`f.Eventually p` or `∀ᶠ x in f, p x` mean that `{x | p x} ∈ f`. E.g., `∀ᶠ x in atTop, p x`
means that `p` holds true for sufficiently large `x`.ProbabilityTheory.mgf : {Ω : Type u_1} → {m : MeasurableSpace Ω} → (Ω → ℝ) → MeasureTheory.Measure Ω → ℝ → ℝMoment-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 Kernel.HasSubexponentialMGF (X : Ω → ℝ) (V b : ℝ)
(κ : Kernel Ω' Ω) (ν : Measure Ω' := by volume_tac) : Prop where
integrable_exp_mul : ∀ t, b * |t| ≤ 1 → Integrable (fun ω ↦ exp (t * X ω)) (κ ∘ₘ ν)
mgf_le : ∀ᵐ ω' ∂ν, ∀ t, b * |t| ≤ 1 → mgf X (κ ω') 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, 60 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.