Probability and Statistic on Infinite-Dimensional spaces

[lokofer]

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:

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

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

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

Or something similar...

 

[Hurkyl]

If you can put a probability measure on it, you can do statistics. No need to "generalize".

 

[lokofer]

The problem is that you can't find any "Infinite dimensional " meassure... unless perhaps that if you have a 1-dimensional meassure you take:

$$\sum_{i}^{\infty} \mu _{i}$$ $$\prod _{i}^{\infty} \mu _{i}$$

sum or product of known meassures... the problem of "probabilistic meassures" for Feynman Path Integral is one of the unsolved problems in Theoretical Physics...

 

[HallsofIvy]

What do you mean 'you can't find any "infinite dimensional" measure'? There are standard measures on Hilbert and Banach spaces.

 

[Hurkyl]

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)