ProbabilityTheory.HasSubexponentialMGF.sum_of_iIndepFun
Lemma
No docstring.
Types
-
Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra. -
ι : Type u_2
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. -
b : ℝ -
X : ι → Ω → ℝ -
V : ι → ℝ -
s : Finset ιFinset αis the type of finite sets of elements ofα.
Assuming
-
h_indep : iIndepFun X μA family of functions defined on the same spaceΩand taking values in possibly different spaces, each with a measurable space structure, is independent if the family of measurable space structures… -
h : ∀ i ∈ s, HasSubexponentialMGF (X i) (V i) b μXhas 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).
Then
HasSubexponentialMGF (∑ i ∈ s, X i) (∑ i ∈ s, V i) b μMeasurableSpace : 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.
Finset : Type u_3 → Type u_3`Finset α` is the type of finite sets of elements of `α`. It is implemented as a multiset (a list up to permutation) which has no duplicate elements.
ProbabilityTheory.iIndepFun : {Ω : Type u_1} →
{ι : Type u_2} →
{_mΩ : MeasurableSpace Ω} →
{β : ι → Type u_6} →
[m : (x : ι) → MeasurableSpace (β x)] →
((x : ι) → Ω → β x) → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.iIndepFun._auto_1 → PropA family of functions defined on the same space `Ω` and taking values in possibly different spaces, each with a measurable space structure, is independent if the family of measurable space structures they generate on `Ω` is independent. For a function `g` with codomain having measurable space structure `m`, the generated measurable space structure is `MeasurableSpace.comap g m`.
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
Code
lemma sum_of_iIndepFun {ι : Type*} {X : ι → Ω → ℝ} (h_indep : iIndepFun X μ) {V : ι → ℝ}
(s : Finset ι) (h : ∀ i ∈ s, HasSubexponentialMGF (X i) (V i) b μ) :
HasSubexponentialMGF (∑ i ∈ s, X i) (∑ i ∈ s, V i) b μProof
by
have : HasSubexponentialMGF (fun ω ↦ ∑ (i : s), X i ω) (∑ (i : s), V i) b μ := by
apply sum_of_iIndepFun_of_forall_aemeasurable
· exact h_indep.precomp Subtype.val_injective
· exact fun i ↦ (h i i.2).aemeasurable
· exact fun i _ ↦ h i i.2
rw [Finset.sum_coe_sort] at this
refine (this.congr (ae_of_all _ fun ω ↦ Finset.sum_attach s (fun i ↦ X i ω))).congr ?_
exact ae_of_all _ fun ω ↦ by simp [Finset.sum_apply]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, 61 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.