Probability and Statistic on Infinite-Dimensional spaces

In summary, the conversation discusses the possibility of generalizing the theories of Probability and Statistic to "Infinite-dimensional" spaces. This includes considering probabilistic phenomena with an infinite number of random variables and defining probabilistic n-th "momentum" of a distribution in the sense of functional integrals. The conversation also explores the idea of using probability measures on infinite-dimensional spaces and discusses the practical challenges of using infinite product measures.
  • #1
lokofer
106
0
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...
 
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  • #2
If you can put a probability measure on it, you can do statistics. No need to "generalize".
 
  • #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 Feynman Path Integral is one of the unsolved problems in Theoretical Physics...
 
  • #4
What do you mean 'you can't find any "infinite dimensional" measure'? There are standard measures on Hilbert and Banach spaces.
 
  • #5
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)
 

1. What is the difference between finite-dimensional and infinite-dimensional spaces in terms of probability and statistics?

Finite-dimensional spaces have a finite number of dimensions, while infinite-dimensional spaces have an infinite number of dimensions. This affects probability and statistics because in finite-dimensional spaces, there is a finite number of possible outcomes, allowing for exact probabilities to be calculated. In infinite-dimensional spaces, there are infinite possible outcomes, making it more challenging to calculate probabilities and requiring different statistical techniques.

2. How is probability and statistics applied in infinite-dimensional spaces?

Infinite-dimensional spaces are commonly used in fields such as functional analysis, stochastic processes, and quantum mechanics. In these fields, probability and statistics are used to analyze and make predictions about systems with infinite dimensions, such as continuous functions or infinite sequences. For example, probability distributions and statistical tests can be used to study the behavior of random processes in infinite-dimensional spaces.

3. What are some common challenges when working with probability and statistics on infinite-dimensional spaces?

One of the main challenges is the lack of a uniform probability measure in infinite-dimensional spaces. In finite-dimensional spaces, the probability of each outcome can be calculated by dividing the number of favorable outcomes by the total number of outcomes. However, in infinite-dimensional spaces, there is no clear notion of "total number of outcomes," making it difficult to define a probability measure. Additionally, the use of certain statistical techniques, such as the central limit theorem, may not be applicable in infinite-dimensional spaces.

4. Can probability and statistics be used to make accurate predictions in infinite-dimensional spaces?

Yes, probability and statistics can be used to make predictions in infinite-dimensional spaces. However, the accuracy of these predictions may be limited due to the challenges mentioned above. In some cases, more sophisticated techniques, such as Bayesian statistics, may be necessary to account for the complexities of infinite-dimensional spaces.

5. How does the concept of convergence play a role in probability and statistics on infinite-dimensional spaces?

Convergence is an important concept in probability and statistics on infinite-dimensional spaces. In particular, the concept of weak convergence, which measures the convergence of probability measures, is commonly used to study the behavior of random variables in infinite-dimensional spaces. Additionally, the notion of almost sure convergence, which guarantees that an event will occur with probability one, is also frequently used in the analysis of infinite-dimensional systems.

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