ProbabilityTheory.HasSubexponentialMGF.measureReal_le_le_exp
Lemma
From the authors
For X, Y two independent random variables with sub-exponential fluctuations such that
μ[X] ≥ μ[Y], the probability that X ≤ Y is bounded by a Bernstein-type term.
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. -
X : Ω → ℝ -
b : ℝ -
Y : Ω → ℝ -
VX : ℝ -
VY : ℝ
Assuming
-
hX : HasSubexponentialMGF (fun ω => X ω - ∫ (x : Ω), X x ∂μ) VX 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). -
hY : HasSubexponentialMGF (fun ω => Y ω - ∫ (x : Ω), Y x ∂μ) VY b μ -
hindep : IndepFun X Y μTwo functions are independent if the two measurable space structures they generate are independent. -
h_le : ∫ (x : Ω), Y x ∂μ ≤ ∫ (x : Ω), X x ∂μ
Then
μ.real {ω | X ω ≤ Y ω} ≤
Real.exp
(-min ((∫ (x : Ω), X x ∂μ - ∫ (x : Ω), Y x ∂μ) ^ 2 / (2 * (VX + VY)))
((∫ (x : Ω), X x ∂μ - ∫ (x : Ω), Y x ∂μ) / (2 * b)))MeasurableSpace : 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.
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
ProbabilityTheory.IndepFun : {Ω : Type u_1} →
{β : Type u_6} →
{γ : Type u_7} →
{_mΩ : MeasurableSpace Ω} →
[MeasurableSpace β] →
[MeasurableSpace γ] →
(Ω → β) → (Ω → γ) → autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.IndepFun._auto_1 → PropTwo functions are independent if the two measurable space structures they generate are independent. For a function `f` with codomain having measurable space structure `m`, the generated measurable space structure is `MeasurableSpace.comap f m`. We use the notation `f ⟂ᵢ[μ] g` for `IndepFun f g μ` (scoped in `ProbabilityTheory`).
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`.
MeasureTheory.Measure.real : {α : Type u_5} → {m : MeasurableSpace α} → MeasureTheory.Measure α → Set α → ℝThe real-valued version of a measure. Maps infinite measure sets to zero. Use as `μ.real s`. The API is developed in `Mathlib/MeasureTheory/Measure/Real.lean`.
Set.ofPred : {α : Type u} → (α → Prop) → Set αTurn a predicate `p : α → Prop` into a set, also written as `{x | p x}`Real.exp : ℝ → ℝThe real exponential function, defined as the real part of the complex exponential
Neg.neg : {α : Type u} → [self : Neg α] → α → α`-a` computes the negative or opposite of `a`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `-` in identifiers is `neg` (when used as a unary operator).
Min.min : {α : Type u} → [self : Min α] → α → α → αReturns the lesser of its two arguments. Conventions for notations in identifiers: * The recommended spelling of `min` in identifiers is `min`. * The recommended spelling of `⊓` in identifiers is `inf` (`⊓` is the preferred notation for `min` when the type is not linearly ordered.).
HPow.hPow : {α : Type u} → {β : Type v} → {γ : outParam (Type w)} → [self : HPow α β γ] → α → β → γ`a ^ b` computes `a` to the power of `b`. The meaning of this notation is type-dependent. Conventions for notations in identifiers: * The recommended spelling of `^` in identifiers is `pow`.
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`.
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`.
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 measureReal_le_le_exp {Y : Ω → ℝ} {VX VY : ℝ}
(hX : HasSubexponentialMGF (fun ω ↦ X ω - μ[X]) VX b μ)
(hY : HasSubexponentialMGF (fun ω ↦ Y ω - μ[Y]) VY b μ)
(hindep : IndepFun X Y μ) (h_le : μ[Y] ≤ μ[X]) :
μ.real {ω | X ω ≤ Y ω}
≤ exp (-min ((μ[X] - μ[Y]) ^ 2 / (2 * (VX + VY))) ((μ[X] - μ[Y]) / (2 * b)))Proof
by
have hXY : HasSubexponentialMGF (fun ω ↦ (Y ω - μ[Y]) - (X ω - μ[X])) (VX + VY) b μ := by
rw [add_comm VX]
refine sub_of_indepFun hY hX ?_
exact hindep.symm.comp (φ := fun x ↦ x - μ[Y]) (ψ := fun x ↦ x - μ[X])
(by fun_prop) (by fun_prop)
calc μ.real {ω | X ω ≤ Y ω}
_ = μ.real {ω | (μ[X] - μ[Y]) ≤ (Y ω - μ[Y]) - (X ω - μ[X])} := by
congr with ω
constructor <;> intro h <;> linarith
_ ≤ exp (-min ((μ[X] - μ[Y]) ^ 2 / (2 * (VX + VY))) ((μ[X] - μ[Y]) / (2 * b))) :=
hXY.measure_ge_le (by linarith)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, 72 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.