ProbabilityTheory.HasSubexponentialMGF.add_of_hasCondSubexponentialMGF
Lemma
From the authors
If X is sub-exponential with parameters (VX, b) with respect to the restriction of μ to
a sub-sigma-algebra m and Y is conditionally sub-exponential with parameters (VY, b) with
respect to m and μ then X + Y is sub-exponential with parameters (VX + VY, b) with
respect to μ.
HasSubexponentialMGF X VX b (μ.trim hm) can be obtained from HasSubexponentialMGF X VX b μ
if X is m-measurable. See HasSubexponentialMGF.trim.
Types
-
Ω : Type u_1m : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.mΩ : MeasurableSpace ΩStandardBorelSpace ΩA standard Borel space is a measurable space arising as the Borel sets of some Polish topology.
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.IsFiniteMeasure μA measureμis called finite ifμ univ < ∞. -
X : Ω → ℝ -
b : ℝ -
Y : Ω → ℝ -
VX : ℝ -
VY : ℝ
Assuming
-
hm : m ≤ mΩ -
hX : HasSubexponentialMGF X VX b (μ.trim hm)Xhas 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). -
hY : HasCondSubexponentialMGF m hm Y VY b μA random variableXhas a conditionally sub-exponential moment-generating function with parameters(V, b)with respect to a sigma-algebramand a measureμif for alltwithb * |t| ≤ 1,…
Then
HasSubexponentialMGF (X + Y) (VX + VY) b μMeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
StandardBorelSpace : (α : Type u_1) → [MeasurableSpace α] → PropA standard Borel space is a measurable space arising as the Borel sets of some Polish topology. This is useful in situations where a space has no natural topology or the natural topology in a space is non-Polish. To endow a standard Borel space `α` with a compatible Polish topology, use `letI := upgradeStandardBorel α`. One can then use `eq_borel_upgradeStandardBorel α` to rewrite the `MeasurableSpace α` instance to `borel α t`, where `t` is the new topology.
MeasureTheory.IsFiniteMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called finite if `μ univ < ∞`.
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.
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
MeasureTheory.Measure.trim : {α : Type u_1} → {m m0 : MeasurableSpace α} → MeasureTheory.Measure α → m ≤ m0 → MeasureTheory.Measure αRestriction of a measure to a sub-σ-algebra. It is common to see a measure `μ` on a measurable space structure `m0` as being also a measure on any `m ≤ m0`. Since measures in mathlib have to be trimmed to the measurable space, `μ` itself cannot be a measure on `m`, hence the definition of `μ.trim hm`. This notion is related to `OuterMeasure.trim`, see the lemma `toOuterMeasure_trim_eq_trim_toOuterMeasure`.
ProbabilityTheory.HasCondSubexponentialMGF : {Ω : Type u_1} →
(m : MeasurableSpace Ω) →
{mΩ : MeasurableSpace Ω} →
m ≤ mΩ →
[StandardBorelSpace Ω] →
(Ω → ℝ) →
ℝ →
ℝ →
(μ : autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasCondSubexponentialMGF._auto_1) →
[MeasureTheory.IsFiniteMeasure μ] → PropA random variable `X` has a conditionally sub-exponential moment-generating function with parameters `(V, b)` with respect to a sigma-algebra `m` and a measure `μ` if for all `t` with `b * |t| ≤ 1`, `exp (t * X)` is `μ`-integrable and the moment-generating function of `X` conditioned on `m` is almost surely bounded by `exp (V * t ^ 2 / 2)`. The actual definition uses `Kernel.HasSubexponentialMGF`: `HasCondSubexponentialMGF` is defined as sub-exponential with respect to the conditional expectation kernel for `m` and the restriction of `μ` to the sigma-algebra `m`.Go to its page
HAdd.hAdd : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HAdd α β γ] → α → β → γ`a + b` computes the sum of `a` and `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `+` in identifiers is `add`.
Code
lemma HasSubexponentialMGF.add_of_hasCondSubexponentialMGF [IsFiniteMeasure μ]
{Y : Ω → ℝ} {VX VY : ℝ} (hm : m ≤ mΩ)
(hX : HasSubexponentialMGF X VX b (μ.trim hm))
(hY : HasCondSubexponentialMGF m hm Y VY b μ) :
HasSubexponentialMGF (X + Y) (VX + VY) b μProof
by
suffices HasSubexponentialMGF (fun p ↦ X p.1 + Y p.2) (VX + VY) b
(@Measure.map Ω (Ω × Ω) mΩ (m.prod mΩ) Function.diag μ) by
have h_eq : X + Y = (fun p ↦ X p.1 + Y p.2) ∘ Function.diag := rfl
rw [h_eq]
refine HasSubexponentialMGF.of_map ?_ this
exact @Measurable.aemeasurable _ _ _ (m.prod mΩ) _ _
((measurable_id'' hm).prodMk measurable_id)
rw [hasSubexponentialMGF_iff_kernel] at hX ⊢
have hY' : Kernel.HasSubexponentialMGF Y VY b (condExpKernel μ m)
(Kernel.const Unit (μ.trim hm) ∘ₘ Measure.dirac ()) := by simpa
convert! hX.add_comp hY'
ext
rw [Kernel.const_apply, ← Measure.compProd, compProd_trim_condExpKernel]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: 3 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.