Talk:Stochastic convergence

Article Checklist for "Stochastic convergence"

Workgroup category or categories	Mathematics Workgroup, Physics Workgroup, Chemistry Workgroup [Categories OK]
Article status	Stub: no more than a few sentences
Underlinked article?	Not specified
Basic cleanup done?	Yes
Checklist last edited by	Ragnar Schroder 11:13, 28 June 2007 (CDT)

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Work in progress

This page is a work in progress, I'm struggling with the latex and some other stuff, there may be actual errors now.

Ragnar Schroder 12:04, 28 June 2007 (CDT)

Almost sure convergence

The definition in the text doesn't make sense. In general, something is true almost surely (a.s.) if it is true with a probability of 1. It is almost surely true that a randomly chosen number is not 4. Greg Woodhouse 12:08, 28 June 2007 (CDT)

Greg, some would argue if the probability has a chance of being not true ≤0.01‰ the convergenge to a value is true, and the tail can be forgotten. Compare to series. Robert Tito |  Talk  12:44, 28 June 2007 (CDT)

I understand. It's a technical concept from measure theory. If two functions (say f and g) are equal except on a set of measure 0, we say f = g almost everywhere. This is important because their (Lebesgue) integrals over a given set will always be equal and it is convenient to identify such functions because then if we define

$(f,g)=\int _{A}f\,{\bar {g}}\,dx$

defines a metric on $\scriptstyle L^{2}(A)$, giving it the structure of a metric space. If we didn't identify functions that wee equal almost everywhere (i.e., treat them as the same function), the purported metric would not be positive definite.

On a foundational level, probability theory is essentially measure theory, (and distributions are just measurable functions that, when integrated ovderf a set A give the probability of A). It is just a convention that probability theorists use the phrase "almost surely" instead of "almost everywhere", the meanings are the same. Of course, I'm almost sure :) you already know this! Greg Woodhouse 13:28, 28 June 2007 (CDT)

Almost sure convergence

I agree with all this, but I think stochastic convergence is a concept that should be accessible to intelligent laymen, not just math/natsci/tech guys.

Therefore, I try hard to avoid reference to hard-core math like measure theory in the beginning of the article, such things should come at the end, after the non-expert has gotten as enlightened as possible wrt the basic ideas.

I've corrected the tex code in the definition. The definition is standard textbook fare, but I really don't like it, I'll try find one that's more intuitive.

Ragnar Schroder 16:30, 28 June 2007 (CDT)

Ragnar, please do, science - specially as encyclopedic science should be available and understandable to as many as possible. Even to the level where analogons are used to visualize a point even when the analogon isn't scientifically correct. If it helps to make laymen understand a topic that is what I would like to call academic freedom in educational and didactical sense. Please continue and make it easier. Robert Tito |  Talk  16:49, 28 June 2007 (CDT)

Yes, the definition is better, but what does it mean? I assume $\scriptstyle X_{i}$ represents a stochastic process of some sort. Are you saying that as an ordinary function of i, the limit is a with probability 1? If it's just a function, why is probability involved? Or are you saying that as $\scriptstyle i\rightarrow \infty$, the distance $|X_{i}-a|$ approaches 0 with probability 1 (whatever that might mean)? The definition still needs to be fleshed out a bit.

On an intuitive level, I think it's clear enough what you mean: the variable $\scriptstyle X_{i}$ represents a "random walk", that gets you closer and closer to a. You don't know where you will be after i steps, but you do know that the probability that you will be any sizeable distance from a becomes vanishingly small as i approaches infinity. How can you translate that into mathematical language? Greg Woodhouse 17:03, 28 June 2007 (CDT)