[Merged by Bors] - feat(probability/strong_law): Lp version of the strong law of large numbers #15392
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JasonKYi
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I disagree that it needs more measurability than the almost sure version: ae measurability is clearly enough for the statement to be true -- you can deduce it from the version where you add measurability, by doing modifications on a measure 0 set, but I'd rather see a direct proof if possible (should be change the definition of uniform integrability to replace |
sgouezel
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I've opened #15623 for this |
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…_integrable` to only require `ae_strongly_measurable` (#15623) The L¹ version of the strong LLN does not require `strongly_measurable` but the assumption in `uniform_integrable` forces it to have this condition if not requiring an extra step to relax the condition (see #15392). This PR relaxes the definition of `uniform_integrable` so it only requires `ae_strongly_measurable`.
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
…r-community/mathlib into JasonKYi/strong_law_L1
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✌️ JasonKYi can now approve this pull request. To approve and merge a pull request, simply reply with |
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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…umbers (#15392) This PR proves the Lp version of the strong law of large numbers which states that $\frac{1}{n}\sum_{i < n} X_i$ converges to $\mathbb{E}[X_0]$ in the Lp-norm where $(X_n)$ is iid. and Lp.
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…_integrable` to only require `ae_strongly_measurable` (leanprover-community#15623) The L¹ version of the strong LLN does not require `strongly_measurable` but the assumption in `uniform_integrable` forces it to have this condition if not requiring an extra step to relax the condition (see leanprover-community#15392). This PR relaxes the definition of `uniform_integrable` so it only requires `ae_strongly_measurable`.
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…umbers (leanprover-community#15392) This PR proves the Lp version of the strong law of large numbers which states that $\frac{1}{n}\sum_{i < n} X_i$ converges to $\mathbb{E}[X_0]$ in the Lp-norm where $(X_n)$ is iid. and Lp.
robertylewis
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…_integrable` to only require `ae_strongly_measurable` (#15623) The L¹ version of the strong LLN does not require `strongly_measurable` but the assumption in `uniform_integrable` forces it to have this condition if not requiring an extra step to relax the condition (see #15392). This PR relaxes the definition of `uniform_integrable` so it only requires `ae_strongly_measurable`.
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…umbers (#15392) This PR proves the Lp version of the strong law of large numbers which states that $\frac{1}{n}\sum_{i < n} X_i$ converges to $\mathbb{E}[X_0]$ in the Lp-norm where $(X_n)$ is iid. and Lp.
khwilson
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…_integrable` to only require `ae_strongly_measurable` (#15623) The L¹ version of the strong LLN does not require `strongly_measurable` but the assumption in `uniform_integrable` forces it to have this condition if not requiring an extra step to relax the condition (see #15392). This PR relaxes the definition of `uniform_integrable` so it only requires `ae_strongly_measurable`.
khwilson
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…umbers (#15392) This PR proves the Lp version of the strong law of large numbers which states that $\frac{1}{n}\sum_{i < n} X_i$ converges to $\mathbb{E}[X_0]$ in the Lp-norm where $(X_n)$ is iid. and Lp.
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This PR proves the Lp version of the strong law of large numbers which states that$\frac{1}{n}\sum_{i < n} X_i$ converges to $\mathbb{E}[X_0]$ in the Lp-norm where $(X_n)$ is iid. and Lp.
uniform_integrableto only requireae_strongly_measurable#15623