ProbabilityTheory.HasCondSubgaussianMGF.hasCondSubexponentialMGF
Lemma
No docstring.
Types
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Ω : Type u_1m : MeasurableSpace ΩA measurable space is a space equipped with a σ-algebra.mΩ : MeasurableSpace ΩStandardBorelSpace ΩA standard Borel space is a measurable space arising as the Borel sets of some Polish topology.
Given
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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.MeasureTheory.IsFiniteMeasure μA measureμis called finite ifμ univ < ∞. -
X : Ω → ℝ -
c : NNReal -
b : ℝ
Assuming
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hm : m ≤ mΩimplicit -
h : HasCondSubgaussianMGF m hm X c μA random variableXhas a conditionally sub-Gaussian moment-generating function with parametercwith respect to a sigma-algebramand a measureμif for allt : ℝ,exp (t * X)isμ-inte…
MeasurableSpace : Type u_6 → Type u_6A measurable space is a space equipped with a σ-algebra.
StandardBorelSpace : (α : Type u_1) → [MeasurableSpace α] → PropA standard Borel space is a measurable space arising as the Borel sets of some Polish topology. This is useful in situations where a space has no natural topology or the natural topology in a space is non-Polish. To endow a standard Borel space `α` with a compatible Polish topology, use `letI := upgradeStandardBorel α`. One can then use `eq_borel_upgradeStandardBorel α` to rewrite the `MeasurableSpace α` instance to `borel α t`, where `t` is the new topology.
MeasureTheory.IsFiniteMeasure : {α : Type u_1} → {m0 : MeasurableSpace α} → MeasureTheory.Measure α → PropA measure `μ` is called finite if `μ univ < ∞`.
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.
NNReal : TypeNonnegative real numbers, denoted as `ℝ≥0` within the NNReal namespace
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`.
ProbabilityTheory.HasCondSubgaussianMGF : {Ω : Type u_1} →
(m : MeasurableSpace Ω) →
{mΩ : MeasurableSpace Ω} →
m ≤ mΩ →
[StandardBorelSpace Ω] →
(Ω → ℝ) →
NNReal →
(μ : autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasCondSubgaussianMGF._auto_1) →
[MeasureTheory.IsFiniteMeasure μ] → PropA random variable `X` has a conditionally sub-Gaussian moment-generating function with parameter `c` with respect to a sigma-algebra `m` and a measure `μ` if for all `t : ℝ`, `exp (t * X)` is `μ`-integrable and the moment-generating function of `X` conditioned on `m` is almost surely bounded by `exp (c * t ^ 2 / 2)` for all `t : ℝ`. This implies in particular that `X` has expectation 0. The actual definition uses `Kernel.HasSubgaussianMGF`: `HasCondSubgaussianMGF` is defined as sub-Gaussian with respect to the conditional expectation kernel for `m` and the restriction of `μ` to the sigma-algebra `m`.
ProbabilityTheory.HasCondSubexponentialMGF : {Ω : Type u_1} →
(m : MeasurableSpace Ω) →
{mΩ : MeasurableSpace Ω} →
m ≤ mΩ →
[StandardBorelSpace Ω] →
(Ω → ℝ) →
ℝ →
ℝ →
(μ : autoParam (MeasureTheory.Measure Ω) ProbabilityTheory.HasCondSubexponentialMGF._auto_1) →
[MeasureTheory.IsFiniteMeasure μ] → PropA random variable `X` has a conditionally sub-exponential moment-generating function with parameters `(V, b)` with respect to a sigma-algebra `m` and a measure `μ` if for all `t` with `b * |t| ≤ 1`, `exp (t * X)` is `μ`-integrable and the moment-generating function of `X` conditioned on `m` is almost surely bounded by `exp (V * t ^ 2 / 2)`. The actual definition uses `Kernel.HasSubexponentialMGF`: `HasCondSubexponentialMGF` is defined as sub-exponential with respect to the conditional expectation kernel for `m` and the restriction of `μ` to the sigma-algebra `m`.Go to its page
Code
lemma _root_.ProbabilityTheory.HasCondSubgaussianMGF.hasCondSubexponentialMGF {c : ℝ≥0}
(h : HasCondSubgaussianMGF m hm X c μ) (b : ℝ) :
HasCondSubexponentialMGF m hm X c b μProof
h.hasSubexponentialMGF b
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: 2 project declarations, 68 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.