`ProbabilityTheory.Kernel.HasSubexponentialMGF.add_compProd`🔗

Lemma

From the authors

For ν : Measure Ω', κ : Kernel Ω' Ω and η : (Ω' × Ω) Ω'', if a random variable X : Ω → ℝ has a sub-exponential mgf with respect to κ and ν and another random variable Y : Ω'' → ℝ has a sub-exponential mgf with respect to η and ν ⊗ₘ κ : Measure (Ω' × Ω), with the same parameter b, then X + Y (random variable on the measurable space Ω × Ω'') has a sub-exponential mgf with respect to κ ⊗ₖ η : Kernel Ω' (Ω × Ω'') and ν.

Types

Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
Ω' : Type u_2mΩ' : MeasurableSpace Ω'
Ω'' : Type u_3mΩ'' : MeasurableSpace Ω''

Given

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.MeasureTheory.SFinite νA measure is called s-finite if it is a countable sum of finite measures.
κ : Kernel Ω' ΩA kernel from a measurable space α to another measurable space β is a measurable function κ : α → Measure β.
X : Ω → ℝ
V : ℝ
b : ℝ
Y : Ω'' → ℝ
VY : ℝ
η : Kernel (Ω' × Ω) Ω''IsZeroOrMarkovKernel ηA class for kernels which are zero or a Markov kernel.

Assuming

hX : HasSubexponentialMGF X V b κ νA 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 int…
hY : HasSubexponentialMGF Y VY b η (ν.compProd κ)

Then

HasSubexponentialMGF (fun p => X p.1 + Y p.2) (V + VY) b (κ.compProd η) ν

MeasurableSpace : Type u_6 → Type u_6

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

MeasureTheory.SFinite : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → Prop

A measure is called s-finite if it is a countable sum of finite measures.

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`.

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)`.

Real : Type

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

ProbabilityTheory.IsZeroOrMarkovKernel : {α : Type u_1} →
  {β : Type u_2} → {mα : MeasurableSpace α} → {mβ : MeasurableSpace β} → ProbabilityTheory.Kernel α β → Prop

A class for kernels which are zero or a Markov kernel.

Prod : Type u → Type v → Type (max u v)

The product type, usually written `α × β`. Product types are also called pair or tuple types.
Elements of this type are pairs in which the first element is an `α` and the second element is a
`β`.

Products nest to the right, so `(x, y, z) : α × β × γ` is equivalent to `(x, (y, z)) : α × (β × γ)`.


Conventions for notations in identifiers:

 * The recommended spelling of `×` in identifiers is `Prod`.

ProbabilityTheory.Kernel.HasSubexponentialMGF : {Ω : Type u_1} →
  {Ω' : Type u_2} →
    {mΩ : MeasurableSpace Ω} →
      {mΩ' : MeasurableSpace Ω'} →
        (Ω → ℝ) →
          ℝ →
            ℝ →
              ProbabilityTheory.Kernel Ω' Ω →
                autoParam (MeasureTheory.Measure Ω') ProbabilityTheory.Kernel.HasSubexponentialMGF._auto_1 → Prop

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`.

Go to its page

MeasureTheory.Measure.compProd : {α : Type u_1} →
  {β : Type u_2} →
    {mα : MeasurableSpace α} →
      {mβ : MeasurableSpace β} → MeasureTheory.Measure α → ProbabilityTheory.Kernel α β → MeasureTheory.Measure (α × β)

The composition-product of a measure and a kernel.

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`.

ProbabilityTheory.Kernel.compProd : {α : Type u_4} →
  {β : Type u_5} →
    {γ : Type u_6} →
      {mα : MeasurableSpace α} →
        {mβ : MeasurableSpace β} →
          {mγ : MeasurableSpace γ} →
            ProbabilityTheory.Kernel α β → ProbabilityTheory.Kernel (α × β) γ → ProbabilityTheory.Kernel α (β × γ)

Composition-Product of kernels. For s-finite kernels, it satisfies
`∫⁻ bc, f bc ∂(compProd κ η a) = ∫⁻ b, ∫⁻ c, f (b, c) ∂(η (a, b)) ∂(κ a)`
(see `ProbabilityTheory.Kernel.lintegral_compProd`).
If either of the kernels is not s-finite, `compProd` is given the junk value 0.

Code

lemma add_compProd {η : Kernel (Ω' × Ω) Ω''} [IsZeroOrMarkovKernel η]
    (hX : HasSubexponentialMGF X V b κ ν) (hY : HasSubexponentialMGF Y VY b η (ν ⊗ₘ κ)) :
    HasSubexponentialMGF (fun p ↦ X p.1 + Y p.2) (V + VY) b (κ ⊗ₖ η) ν

Proof

by
  by_cases hκ : IsSFiniteKernel κ
  swap; · simp [hκ]
  refine .of_ae_mgf_le (fun t ht ↦ integrable_exp_add_compProd hX hY ht) fun t ht ↦ ?_
  filter_upwards [hX.mgf_le, hX.ae_integrable_exp_mul ht, Measure.ae_ae_of_ae_compProd hY.mgf_le,
    Measure.ae_integrable_of_integrable_comp <| integrable_exp_add_compProd hX hY ht]
    with ω' hX_mgf hX_int hY_mgf h_int_mul
  calc mgf (fun p ↦ X p.1 + Y p.2) ((κ ⊗ₖ η) ω') t
  _ = ∫ x, exp (t * X x) * ∫ y, exp (t * Y y) ∂(η (ω', x)) ∂(κ ω') := by
    simp_rw [mgf, mul_add, exp_add] at h_int_mul ⊢
    simp_rw [integral_compProd h_int_mul, integral_const_mul]
  _ ≤ ∫ x, exp (t * X x) * exp (VY * t ^ 2 / 2) ∂(κ ω') := by
    refine integral_mono_of_nonneg ?_ (hX_int.mul_const _) ?_
    · exact ae_of_all _ fun ω ↦ mul_nonneg (by positivity)
        (integral_nonneg (fun _ ↦ by positivity))
    · filter_upwards [hY_mgf] with ω hY_mgf
      gcongr
      exact hY_mgf t ht
  _ ≤ exp ((V + VY) * t ^ 2 / 2) := by
    rw [integral_mul_const, add_mul, add_div, exp_add]
    gcongr
    exact hX_mgf t ht

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, 71 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.Kernel.HasSubexponentialMGF.add_compProd🔗

`ProbabilityTheory.Kernel.HasSubexponentialMGF.add_compProd`🔗