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nLab > Latest Changes: central limit theorem

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2019

    Page created, but author did not leave any comments.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2019

    started a bare minimu, for the moment just so as to make links work

  3. Putting in a rough idea for a community diagram to represent this. More work is needed.

    John Ericson

    diff, v2, current

  4. Better to label morphisms more and objects less

    John Ericson

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTime4 days ago

    What is g ?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTime4 days ago
    • (edited 4 days ago)

    hereby moving John Ericson’s suggestion from a (deprecated) query box environment in the entry to here:

    Perhaps a diagram like could be used to express the central limit theorem:

    Ω ⟶f 𝒩 sample↓ ↓sample X n ⟶g Y n \begin{array}{ccc} \Omega &\overset{f}{\longrightarrow}& \mathcal{N} \\ \mathllap{^{sample}}\Big\downarrow && \Big\downarrow\mathrlap{^{sample}} \\ X_n &\underset{g}{\longrightarrow}& Y_n \end{array}

    where

    • f(Ω)=𝒩(μ Ω,σ Ω 2)f(\Omega) = \mathcal{N}\left(\mu_\Omega, \sigma^2_\Omega\right)

      maps a distribution with finite mean and variance to the normal distribution with the same mean and variance.

    • g(X)=n↦μ+∑ m=0 nX m−μng(X) = n \mapsto \mu + \frac {\sum_{m = 0}^n X_m - \mu} {\sqrt{n}}

      maps one (random) sequence to another

    However, it needs some fixing up. A major issue is the transformed sequence merely converges like a random sequence sampled from the normal distribution. Random sequences are hard to formalize for good reason.

    (A definition of gg is offered right there, but I am not sure if it quite parses.)

    diff, v4, current

    • CommentRowNumber7.
    • CommentAuthorEricson2314
    • CommentTime1 day ago

    Ah, I knew I was seeing fewer query boxes than I used to see, but didn’t realize the rules changed. Sorry about that!

    A definition of g is offered right there, but I am not sure if it quite parses.

    Oh, that was suppose to be the most standard/on-solid-ground part! The formula can be more or less gotten from Wikipedia or https://terrytao.wordpress.com/2010/01/05/254a-notes-2-the-central-limit-theorem/

    g(X) n=μ+∑ m=1 nX m−μn g(X)_n = \mu + \frac {\sum_{m=1}^{n} X_m - \mu} {\sqrt n}

    Is that better? (I switched to m=1m = 1 because I realized Tao was doing 1-based sequence indexing.)

    I realized there is perhaps a bigger diagram that could work for this. There is sort of 3 equivalent transformations that can be done.

    The first is the top morphism, just get the mean and variance, and construct the normal distribution with that mean and variance.

    The second (new) the limit of the sequence I made. The limit of convolving the distribution in the right way (what one does with the probability density functions for adding the independent random variables) is the same normal distribution.

    And then third is path through left and bottom morphisms, the hand-wavey “sample” multiple times, and then transform the sequence with the “fold”. Exactly how the transformed sequence relates to the normal distribution is a bit fuzzy — but how it relates to each intermediate distribution underneath the limit is nice. Also, this is nice and constructive, in that we know transforming sequences of numbers with a fold like this is nice coinduction, and you don’t need to know the underlying PDF or mean or variance of the distribution.

    • CommentRowNumber8.
    • CommentAuthorPaoloPerrone
    • CommentTime1 day ago

    In case it can help, there are categorical formalizations of some random sequences, for example iid random variables.

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