The paper attached below on page 35 says:
...."Kolmogorov existence of the stochastic process describing all P(A) in a coupled way subjects these
to consistency the requirements expressing that if A1;A2 partition A, then P(A1) + P(A2) must have the same distribution as P(A)" Is this extendible to the countable infinite case, with regard to limits of frequency distributions, and if so, or if does under certain conditions, is there a name for this theorem. Is this Kolmogorov's extension theorem.
I know that there are issues for frequency convergence in countably infinite sequences, for example with regard to the space not being a field, that is apart from issues of countable additvity, 'that is apart from 'where the tautology has probability one, but each of its disjuncts, whilst possible has probability zero, and so the sum of the disjuncts has probability zero);
For example even if we only apply finite additivity, there are certain cases where the limiting relative frequency of disjunction is defined, but the disjuncts are not, and the limiting relative frequency of a conjunction is defined but the conjuncts are not; not that they have different values, but that the disjunction has a value, and the disjunct do not have well defined value, ie do not converge at all. I am not sure if it goes the other way, where the disjuncts have well defined limiting relative frequencies but where addivity fails not by way of the disjunction having a different limiting relative frequency, but instead because it does not converge, or rather have a defined limiting relative frequency at all.
This or rather the second paper attached, also speaks or hints about stronger limit theorems, insofar as it mentions the connection to frequentism, hyper-rectangle, tail-free priors, but are all of these
second order/martingale, super-martingale convergence results ,
prequentist (Dawid, Vovk, Schafer) convergence of opinion/frequency results,
the theorems of some non-standard (analysis) probability models ith their 'exact strong laws of large numbers,,
row column models (hoover, third article attached) w (Keisler, hoover, Nelson, Perkins),
Doob's theorem (almost sure consistency theorem), sobolev ball, hyperrectangle models, urn/polya urn models, which are mentioned in the second paper and other uniform convergence,and/ or other stronger relative frequency limit theorems,Kolmogorov couplings, 'dominated law of large numbers' able to get rid of the almost surety, in
(1) place of certainty, or to )
2_replace the measure theoretic 'almost sure' with not only a combinatorial 'almost sure' as Pollock's model in the 1980's (on the problem of induction, probability and epistemology 1983 i think) can, but
(3)an empirical combinatorial 'almost sure', ie instead of
(A) 'almost all possible combinations of infinite outcome sequences,( that a singular infinite sequence may give), give the right relative frequency,
(B) rather an ensemble like result, which elicits a 'factual almost all'; almost all distinct infinite sequences, will necessarilly, if trialled, lead to an infinite sequence whose outcomes match the probability valuesetc, which is much stronger, as it quantifys over distinct triallings. (2) does not entail (3) even if one trialled as many (or more) infinite sequences, then the cardinality (uncountable) over which the 'almost all' in (2) quantifies (the set of all infinite possible infinite outcome sequences.
(2)quantifies over possible outcomes, of the same trial; the sequences are only distinct in virtue of the distinct outcomes, but are nonetheless conceived as the same trial. (2) does not, unlike (3) quantify over distinct triallings/ distinct infinite sequences, that are qualitative similar (identifically distributed) but which are already distinct, independently of whether they have distinct outcomes or not), whereupon it is suggested that almost all actual infinite (distinct but identically distributed) trial sequences, if trialled together, will elicit such an relative frequencies infinite sequences. Frequency limits, and is the following a theorem for countable infinite frequency distributiions?
Clearly one cannot use the result in (2) to get (3) as that would be suggest that one is repeating the same un-repeatable trial, and that (2) that the very same trial has or could have multiple inconsistent outcomes
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability