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<Stochastic analysis thread>
Question:
Is convergence in probability
[math] { plim }_{ n \to \infty } X_n = X [/math]
formally given by
[math] \forall( \varepsilon > 0).\ \lim_{n \to \infty } Pr \left (d(X_n, X) \geq \varepsilon \right) = 0 [/math]
?
And does anyone know Hairers work? I only read one of his paper last year when he got the price, and I get the gist of it - he does a sort of Tailor expansions for super non-smooth objects.
Anyone here working with that?
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Yes.
No.
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>>7806748
Why do you post a first year undergrad definition and then a research question? Are you trying to bait?
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>>7806812
The definition is what I came up with reading a plane text description of the concept and the question about Hairer is because I just stalked him on MathOverflow and was thinking what he's up to
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>>7806835
ad.: in particular I feel fuzzy about the inner normal limit
lim Pr(...)=0
here surely they mean the limit in the the basic norm over R, so this sorta becomes a proper sub-subject of analysis
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>>7806856
What else? Probabilities are real numbers in [0,1].
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>>7806863
I think I was hoping the limit in probability would do without being reducible to the standard notion of limit that requires a metric.
The latter is in principle arbitrary and now the concept of probability (which I think isn't well understood - the general informal notion, not the mathematical theories trying to formalize it) depends on it.
It seems to imply it'll be hard to change perspective to more ad hoc theories for problems eventually involving stuff like stochastic integration.
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>>7806873
Not sure if this is really a mathematical problem or just autistic philosocuck talk., Could you formally explain what you want?
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>>7806874
it's both
if the theory of probability depends on something so specific and complicated like the standard metric on R (lots and lots of math to set it up, as compared to group theory, say), then there's probably no more direct route to solving problems involving probability/stochastic. E.g. if you write software that's supposed to work on a fairly algebraic and logically deductive framework. If plim depends on lim, you might not be able to get around numerics (as opposed to the schematic "thinking" that a compiler does to do its job)
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>>7806893
Well you do have algebraic rules for limits in probability, e.g. that they are preserved by continuous mappings. Also in most cases you prove stochastic convergence by showing something stronger, like Lp-convergence or almost sure convergence.
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