Definition talk:Almost Sure Convergence

What is the point of the following complicated and unreasonable expression?

$\forall \epsilon \in \R_{>0}: \ds \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} < \epsilon} = 1$

Is this a mistake?

The definition should be:

$\forall \epsilon \in \R_{>0}: \ds \map \Pr {\limsup_{n \mathop \to \infty} \size {X_n - X} < \epsilon} = 1$

Or, more simply:

$\ds \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} = 0 } = 1$

Can someone please check the source? --Usagiop (talk) 22:06, 19 September 2023 (UTC)

I'm interested: why is $\forall \epsilon \in \R_{>0}: \ds \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} < \epsilon} = 1$ "complicated and unreasonable", while the one you suggest in its place looks just the same except with an extra 3 letters in it?

Sorry, but I don't know my way around this area in any detail, and never grasped the subtleties. --prime mover (talk) 22:11, 19 September 2023 (UTC)

I wrote this back in high school and probably would not write it this way now. The way it appears here is exactly the way it appears in the source text. I think the confusion arises from writing $\lim$ when a limit is not known to exist. But I think when the authors write the event:

$\ds \set {\lim_{n \mathop \to \infty} \size {X_n - X} < \epsilon}$

strictly speaking they mean:

$\ds \set {\omega \in \Omega : \lim_{n \mathop \to \infty} \size {\map {X_n} \omega - \map X \omega} \text { exists and is } < \epsilon}$

Taking the intersection over $\epsilon > 0$ in the definition then gives:

$\ds \map \Pr {\lim_{n \mathop \to \infty} \size {X_n - X} = 0 } = 1$

and the converse is obvious. $\limsup$ is fine as well, if the $\limsup$ of a non-negative sequence is zero then the sequence must converge (just by writing out what this means) and averts the issue of having to say "the limit exists and [...]". I think in practice I would use none of these, instead I would look at events like $\set {\cmod {X_n - X} \ge \epsilon_n \text { i.o.} }$ for some $\epsilon_n \to 0$ and use some kind of Borel-Cantelli argument to show this has probability $0$, which implies almost sure convergence. Caliburn (talk) 22:36, 19 September 2023 (UTC)

prime mover, you can safely ignore $\Pr$.

I am saying that:

$\forall \epsilon \in \R_{>0}: \ds \lim_{n \mathop \to \infty} \size {X_n - X} < \epsilon$

is nothing but:

$\lim_{n \mathop \to \infty} \size {X_n - X} = 0$

What is the point of using $\epsilon > 0$ this way? The first expression is formally more complex because you need to check first that the limit exists and then it is smaller than all $\epsilon >0$.

However, it could be a little advantage to say instead:

$\forall \epsilon \in \R_{>0}: \ds \limsup_{n \mathop \to \infty} \size {X_n - X} < \epsilon$

But it is OK if this is the the source text. --Usagiop (talk) 22:53, 19 September 2023 (UTC)

If you can find a source which has a "better" way to implement this definition, then feel free to do so, and craft the appropriate equivalence proof.

Till then we revert to our house policy that we implement a definition page as and when we find a compliant source which documents it. --prime mover (talk) 05:14, 20 September 2023 (UTC)