ProbabilityTheory.HasSubexponentialMGF.add_of_holderConjugate
From the authors
Sum of two (not necessarily independent) sub-exponential variables, by Hölder's inequality
with conjugate exponents p and q. The case p = q = 2 is add.
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Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.
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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. -
X : Ω → ℝ -
Y : Ω → ℝ -
VX : ℝ -
VY : ℝ -
bX : ℝ -
bY : ℝ -
p : ℝ -
q : ℝ
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hpq : p.HolderConjugate qReal numbersp q : ℝare **Hölder conjugate** if they are positive and satisfy the equalityp⁻¹ + q⁻¹ = 1. -
hX : HasSubexponentialMGF X VX bX 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). -
hY : HasSubexponentialMGF Y VY bY μ
HasSubexponentialMGF (fun ω => X ω + Y ω) (p * VX + q * VY) (max (p * bX) (q * bY)) 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.
Real.HolderConjugate : ℝ → ℝ → PropReal numbers `p q : ℝ` are **Hölder conjugate** if they are positive and satisfy the equality `p⁻¹ + q⁻¹ = 1`. This is an abbreviation for `Real.HolderTriple p q 1`. This condition shows up in many theorems in analysis, notably related to `L^p` norms. It is equivalent that `1 < p` and `p⁻¹ + q⁻¹ = 1`. See `Real.holderConjugate_iff`.
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
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`.
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`.
Max.max : {α : Type u} → [self : Max α] → α → α → αReturns the greater of its two arguments. Conventions for notations in identifiers: * The recommended spelling of `max` in identifiers is `max`. * The recommended spelling of `⊔` in identifiers is `sup` (`⊔` is the preferred notation for `max` when the type is not linearly ordered.).
Code
lemma add_of_holderConjugate {Y : Ω → ℝ} {VX VY bX bY p q : ℝ} (hpq : p.HolderConjugate q)
(hX : HasSubexponentialMGF X VX bX μ) (hY : HasSubexponentialMGF Y VY bY μ) :
HasSubexponentialMGF (fun ω ↦ X ω + Y ω) (p * VX + q * VY) (max (p * bX) (q * bY)) μProof
by rw [hasSubexponentialMGF_iff_kernel] at hX hY ⊢ exact hX.add_of_holderConjugate hpq hY
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, 58 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.