ProbabilityTheory.HasSubexponentialMGF.measure_abs_average_ge_le
Lemma
From the authors
Tail bound for an average of independent sub-exponential variables with parameters
(V i, b): writing K for the number of variables and V̄ := (∑ i, V i) / K,
P(|(∑ i, X i) / K| ≥ t) ≤ 2 exp (-K min (t ^ 2 / (2 V̄)) (t / (2 b))).
Types
-
Ω : Type u_1mΩ : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra. -
ι : Type u_2Fintype ιFintype αmeans thatαis finite, i.e. there are only finitely many distinct elements of typeα.Nonempty ι
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 -
t : ℝ
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 : ι), 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). -
ht : 0 ≤ t
Then
μ.real {ω | t ≤ |(∑ i, X i ω) / ↑(Fintype.card ι)|} ≤
2 * Real.exp (-(↑(Fintype.card ι) * min (t ^ 2 / (2 * ((∑ i, V i) / ↑(Fintype.card ι)))) (t / (2 * b))))MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
Fintype : Type u_3 → Type u_3`Fintype α` means that `α` is finite, i.e. there are only finitely many distinct elements of type `α`. The evidence of this is a finset `elems` (a list up to permutation without duplicates), together with a proof that everything of type `α` is in the list.
Nonempty : Sort u → Prop`Nonempty α` is a typeclass that says that `α` is not an empty type, that is, there exists an element in the type. It differs from `Inhabited α` in that `Nonempty α` is a `Prop`, which means that it does not actually carry an element of `α`, only a proof that *there exists* such an element. Given `Nonempty α`, you can construct an element of `α` *nonconstructively* using `Classical.choice`.
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.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
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}`abs : {α : Type u_1} → [Lattice α] → [AddGroup α] → α → α`abs a`, denoted `|a|`, is the absolute value of `a`
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`.
Fintype.card : (α : Type u_3) → [Fintype α] → ℕ`card α` is the number of elements in `α`, defined when `α` is a fintype.
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`.
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`.
Code
lemma measure_abs_average_ge_le {ι : Type*} [Fintype ι] [Nonempty ι] {X : ι → Ω → ℝ}
(h_indep : iIndepFun X μ) {V : ι → ℝ} (h : ∀ i, HasSubexponentialMGF (X i) (V i) b μ)
{t : ℝ} (ht : 0 ≤ t) :
μ.real {ω | t ≤ |(∑ i, X i ω) / Fintype.card ι|}
≤ 2 * exp (-(Fintype.card ι
* min (t ^ 2 / (2 * ((∑ i, V i) / Fintype.card ι))) (t / (2 * b))))Proof
by
have hK : (0 : ℝ) < Fintype.card ι := by
have := Fintype.card_pos (α := ι)
positivity
have hbound := (average_of_iIndepFun h_indep h).measure_abs_ge_le ht
refine hbound.trans (le_of_eq ?_)
congr 3
rw [(monotone_mul_left_of_nonneg hK.le).map_min]
congr 1
· field_simp
· field_simpMeaning 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, 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.