Talk:Stochastic convergence

Article Checklist for "Stochastic convergence"

Workgroup category or categories	Mathematics Workgroup, Physics Workgroup, Chemistry Workgroup [Categories OK]
Article status	Developing article: beyond a stub, but incomplete
Underlinked article?	No
Basic cleanup done?	Yes
Checklist last edited by	Greg Woodhouse 20:33, 28 June 2007 (CDT), 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.

Yes, but this is the talk page. :-) Greg Woodhouse 17:05, 28 June 2007 (CDT)

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?

That's what I mean when I say I don't like the compact expression - it's too confusing, especially when the limit itself is a stochastic variable. The formal definition used in my old textbook is $P(\lim _{i\to \infty }X_{i}=X)=1$. Grad students may enjoy getting something like that thrown at them, but not a general audience. I hope the definition section is clear now.

Ragnar Schroder 14:04, 3 July 2007 (CDT)

That makes more sense. It isn't actually stated that a is a stochastic process. I'll be honest: When I firswt looked at this, I thought: "What does this mean? It looks like you're saying that a sequence of stochastic processes converges to a number?" Greg Woodhouse 14:38, 3 July 2007 (CDT)

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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)

You focus on the convergence in probability implication of a.s. convergence here. I think a detailed discussion of the relation between the two should be mentioned after the introduction to the various modes of convergence, in a new subsection to the /* Relations between the different modes of convergence */ section.

Ragnar Schroder 14:04, 3 July 2007 (CDT)

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I don't understand this fragment:

"The number a may also be the outcome of a stochastic variable X. In that case the compact notation $P(\lim _{i\to \infty }X_{i}=X)=1$ is often used."

Isn't it the case that if a stochastic variable converges almost surely, the limit is a deterministic constant and not a stochastic variable? That is, $P(\lim _{i\to \infty }X_{i}=X)=1$ can only be true if X takes some value with probability one and is thus not really a stochastic variable. Or do you implicitly allow for the possibility that X depends on the {X_i} ? -- Jitse Niesen 10:13, 5 July 2007 (CDT)

I'm glad I'm not the only one that finds this notation confusing. Unless, I'm wrong, this is just another way of saying that a sequence of measurable functions converges to a measurable function X a.e. (pointwise), convergence in probability is just convergence in measure, and convergence in the pth order mean is just L^p convergence. The strong law of large numbers (the hypotheses of which I can look up when I get home) ensures convergence a.e. (I think!). I also think(!) that it appliews to i.i.d. random variables, and, in particular to Bernoulli trials. But I'm a little out of my league at this level of probability theory. Greg Woodhouse 11:20, 5 July 2007 (CDT)

The compact notation $P(\lim _{i\to \infty }X_{i}=X)=1$ *is* confusing. Unfortunately, it seems to be traditional, so I guess it has to be included, even though a small rather than cap X would be more intuitive. Think of it this way: The value of X is obtained in advance, f.i. by throwing a die. Then somehow the sequence converges as to that value.

However, this is a simplification - the value of X may be unknown in advance, and the sequence $X_{i}$ then basically may tell us more and more what the outcome of X would be, had we performed that particular random experiment.

I'll try to think of a simple, concrete example to illustrate this.

Ragnar Schroder 11:14, 5 July 2007 (CDT)

examples?

Can anyone think of a non-trivial example of a.s. convergence (i.e., one where the sequence doesn't eventually become constant)? Greg Woodhouse 20:59, 28 June 2007 (CDT)

many chemical processes and descriptions thereof are using stochastics I will have to see if I can pop up with a decent example. But as example (top of my head) it can be used to derive much of physical chemistry. Robert Tito |  Talk 

Well, I did a bit of looking and the strong law of large numbers has as its conlusion a.s. convergence. (I didn't know that. I was aware of the weak law of large numbers, but never bothered to look up the strong version.) Anyway, I added this as a third example. Greg Woodhouse 21:52, 28 June 2007 (CDT)

An example of non-trivial a.s. convergence:

Assume a guy has two sources of income, one stochastic. On a given day j he gets $$X_{j}$ from the stochastic one, and $$f(j)$ from the other. Assume the first one converges almost surely to 0. Then obviously his total income $U_{j}=X_{j}+f(t)$ converges almost surely to $f(t)$.

Ragnar Schroder 23:20, 28 June 2007 (CDT)

I've deleted example 3. Greg Woodhouse 23:38, 28 June 2007 (CDT)

Categories

I'm uncertain about the categories for this article, it's now listed under Mathematics, Physics and Chemistry. AFAIK, Stochastic Differential Equations is used a lot in Economics, maybe more than in f.i. chemistry???

Other areas include computer science,operations research and mathematical biology. If we tried to include every are where these ideas can be applied, the list could become a bit unwieldy! On the other hand, that doesn't mean we shouldn't include certain core applications areas, there really are no hard and fast rules. Greg Woodhouse 20:52, 28 June 2007 (CDT)

Relationships between the types of convergence

I commented out the last claim in this section because I believe it is false as stated. Perhaps you had something different in mind here. Greg Woodhouse 21:34, 28 June 2007 (CDT)

Oops...I've got mud on my face here! Looking a bit more closely, I see that you're saying that L^p convergence implies convergence in measure. This is true. Greg Woodhouse 21:44, 28 June 2007 (CDT)

Added example 3 is under wrong convergence

Example 3 of the Almost sure convergence section should be moved to the Convergence in probability section.

Rather surprisingly, the sequence does NOT converge a.s, only P, AFAIK.

Ragnar Schroder 22:58, 28 June 2007 (CDT)

Ok. It DOES converge a.s. as well as P. A.s. convergence follows from the "Strong law of large numbers", P convergence from Khinchin's theorem, aka the "weak law of large numbers".

I think the example was very good, and should be put back either where it was, or under convergence in probability.

Ragnar Schroder 22:09, 30 June 2007 (CDT)