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Probability and Statistic on Infinite-Dimensional spaces

  1. Sep 1, 2006 #1
    Probability and Statistic on "Infinite-Dimensional" spaces

    Hello..can the theories of Probability and Statistic be generalized to "Infinite-dimensional" spaces?..i mean if there are "probabilistic" phenomenon that include an infinite number of random variables, or include "random functions" instead of random numbers, or if you can define the probabilistic n-th "momentum" of a distribution in the sense of the functional integral:

    [tex] \int D[\phi ]\phi^{n} P[\phi]= < \phi ^{n} > [/tex]

    By the way..if Montecarlo integration does not depend on the dimensionality of space..:grumpy: why can't you perform infinite dimensional integrals...? simply in the form:

    [tex] \int D[\phi ]\phi^{n} P[\phi]= \sum_{i} P[ \phi _i ] \phi_{i}^{n} + \sum_{r}a(r) \delta ^{r}\phi^{n} P[\phi] [/tex]

    Or something similar...
     
    Last edited: Sep 1, 2006
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  3. Sep 1, 2006 #2

    Hurkyl

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    If you can put a probability measure on it, you can do statistics. No need to "generalize".
     
  4. Sep 1, 2006 #3
    The problem is that you can't find any "Infinite dimensional " meassure... unless perhaps that if you have a 1-dimensional meassure you take:

    [tex] \sum_{i}^{\infty} \mu _{i} [/tex] [tex] \prod _{i}^{\infty} \mu _{i} [/tex]

    sum or product of known meassures... the problem of "probabilistic meassures" for Feynmann Path Integral is one of the unsolved problems in Theoretical Physics....
     
  5. Sep 2, 2006 #4

    HallsofIvy

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    What do you mean 'you can't find any "infinite dimensional" measure'? There are standard measures on Hilbert and Banach spaces.
     
  6. Sep 2, 2006 #5

    Hurkyl

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    I don't see any theoretical problem with taking an infinite product measure. (that doesn't mean none exists...) There is a practical problem, though -- too many interesting sets have infinite measure, or zero measure. E.g. the measure of a cube is:

    0 (if the side length is less than 1)
    1 (if the side length equals 1)
    +infinity (if the side length is greater than 1)
     
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