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As I suggested in a previous questions, is it merely the case that all of these stronger probability to limiting relative frequency theorems, are only stronger in the sense, (A) not (B)
(A)that the criterion that must be satisfied for convergence, are relaxed under which almost sure convergence occurs, or the speed of such convergence is faster then in conventional probability spaces (via the strong law of large numbers);
Such as,
ie instead of the trials having to be independently distributed, they need only be exchangeable or partially exchangeable sequences, or neither
- or that they can deal with situations where only finite additivity holds (where the strong law of large numbers needs countable addirvity), or
-where the variance is greater or infinite.
-or that almost sure fixed point convergence is faster (and or uniform, unlike in the case of equi-probable events, (where by the law of the iterated algorithm convergence is much slower for these kinds of events)
-or that in some weird sense, the almost sure, is an even bigger Pr(1) but not a certainty, and or the proportion or measure of sequences whose absolute frequency difference from the probability value is also much greater, not just the relative frequency difference
(B) that convergence is certain, ie pointwise and absolute; or that 'almost all distinct countable infinite sequences of distinct but identically distributed trials, in a collective of countably infinitely many such sequences converge to a limit. The limiting relative frequency of sequences, in the infinite sequence of infinite sequences converge, to a limit
Feller (1968 volume 2) cites some interesting examples > of apparently stronger results although not of this form, ladder epochs, karamaters regular variations, and infinite convolutions but nothing so strong as absolute
Marshall second attached makes reference to book by engel 1992 'the road to randomness' where marshall (second attached) says on page 347 that engel proves some results about relative frequency to probabilityconvergence, (within a limited context), that are not qualified by probabilistic notions (such almost surely) but I had difficulty finding such notions
I know there are feller process; i heard that in banach spaces, they have a more combinatorial flavour; ie these uniform limit theorems and the like. I think one such text suggests that all but countably many sequences will converge, although i dont kow what kind of combination or measure is used for that- ie is the cardinality of the measure of the non limiting, degenerate sequence only couintable in comparison with regard to the correct sequence, which would mean they have to define the measure i should on the real line not just on [0,1] and do they mean all but countably many outcome sequence or all but countably many trial sequence; either way something is going on that is strongter, which somehow leaves out certain sequences if they are speaking about all but countably many if it in this combinatorial sense. I presume countably infinitely many, but countable nonetheless
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