Theorems · Theorem · probability

MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto

∀ {Ω : Type u_2} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0}
  {f : ℕ → Ω → ℝ} {R : NNReal} [MeasureTheory.IsFiniteMeasure μ],
  MeasureTheory.Submartingale f ℱ μ →
    (∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R) →
      ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Filter.Tendsto (fun n => f n ω) Filter.atTop (nhds c)

One-sided martingale bound: If f is a submartingale which has uniformly bounded differences, then for almost every ω, f n ω is bounded above (in n) if and only if it converges.

Defined in: Mathlib.Probability.Martingale.BorelCantelli
Cited by: 1 results in Mathlib
Foundations: Depth 319 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: MeasureTheory.IsFiniteMeasure

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Cites36

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement and proof · cited by 25,697
MeasurableSpacestatement and proof · cited by 13,106
MeasureTheory.Measurestatement and proof · cited by 10,939
nhdsstatement and proof · cited by 5,554
Set.rangestatement and proof · cited by 4,705
NNRealstatement and proof · cited by 4,310
Filter.Tendstostatement and proof · cited by 3,814
Filter.Eventuallystatement and proof · cited by 3,134
Filter.atTopstatement and proof · cited by 2,405
MeasureTheory.aestatement and proof · cited by 2,352
absstatement and proof · cited by 1,814
Filter.univ_mem'proof · cited by 1,672

Cited by1

Results whose statement or proof uses this declaration.

MeasureTheory.Martingale.bddAbove_range_iff_bddBelow_rangeproof · cited by 2