ProbabilityTheory.hasSubexponentialMGF_sub_integral_of_abs_le
Lemma
From the authors
If |X| ≤ M almost surely, then X - μ[X] has a sub-exponential mgf with parameters
(3 * Var[X] / 2, 2 * M).
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.MeasureTheory.IsProbabilityMeasure μA measureμis called a probability measure ifμ univ = 1. -
X : Ω → ℝ -
M : ℝ
Assuming
-
hm : AEMeasurable X μA function is almost everywhere measurable if it coincides almost everywhere with a measurable function. -
hM : ∀ᵐ (ω : Ω) ∂μ, |X ω| ≤ Mf.Eventually por∀ᶠ x in f, p xmean that{x | p x} ∈ f.
Then
HasSubexponentialMGF (fun ω => X ω - ∫ (x : Ω), X x ∂μ) (3 * variance X μ / 2) (2 * M) μX 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).MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
MeasureTheory.IsProbabilityMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called a probability measure if `μ univ = 1`.
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.
AEMeasurable : {α : Type u_1} →
{β : Type u_2} →
[MeasurableSpace β] →
{_m : MeasurableSpace α} → (α → β) → autoParam (MeasureTheory.Measure α) AEMeasurable._auto_1 → PropA function is almost everywhere measurable if it coincides almost everywhere with a measurable function. A similar notion is `MeasureTheory.NullMeasurable`. That notion is equivalent to `AEMeasurable` if the σ-algebra on the codomain is countably generated, but weaker in general.
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`.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`.
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
HSub.hSub : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HSub α β γ] → α → β → γ`a - b` computes the difference of `a` and `b`. The meaning of this notation is type-dependent. * For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. Conventions for notations in identifiers: * The recommended spelling of `-` in identifiers is `sub` (when used as a binary operator).
MeasureTheory.integral : {α : Type u_6} →
{G : Type u_7} →
[inst : NormedAddCommGroup G] → [NormedSpace ℝ G] → {x : MeasurableSpace α} → MeasureTheory.Measure α → (α → G) → GThe Bochner integral
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`.
ProbabilityTheory.variance : {Ω : Type u_1} → {mΩ : MeasurableSpace Ω} → (Ω → ℝ) → MeasureTheory.Measure Ω → ℝThe `ℝ`-valued variance of a real-valued random variable defined by applying `ENNReal.toReal` to `evariance`.
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
lemma hasSubexponentialMGF_sub_integral_of_abs_le [IsProbabilityMeasure μ] {M : ℝ}
(hm : AEMeasurable X μ) (hM : ∀ᵐ ω ∂μ, |X ω| ≤ M) :
HasSubexponentialMGF (fun ω ↦ X ω - μ[X]) (3 * Var[X; μ] / 2) (2 * M) μProof
by
have hX : Integrable X μ :=
Integrable.of_mem_Icc (-M) M hm (hM.mono fun ω h ↦ abs_le.1 h)
have hEX : |μ[X]| ≤ M := by
refine (abs_integral_le_integral_abs).trans ?_
calc ∫ ω, |X ω| ∂μ ≤ ∫ _, M ∂μ := integral_mono_ae hX.abs (integrable_const _) hM
_ = M := by simp
refine hasSubexponentialMGF_of_abs_le_of_integral_eq_zero (hm.sub_const _) ?_ ?_ ?_
· filter_upwards [hM] with ω hω
calc |X ω - μ[X]| ≤ |X ω| + |μ[X]| := abs_sub _ _
_ ≤ M + M := add_le_add hω hEX
_ = 2 * M := by ring
· simp [integral_sub hX (integrable_const _)]
· simp [variance_eq_integral hm]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, 69 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.