# strong law of large numbers with a (normal) distribution based visualization (reframed)

To clarify the title, I've been trying to visualize the two versions (weak and strong) of the law of large numbers.

The weak law in the context of the law of large numbers I've undestood in the following sense:

For a large enough n, the distribution of (sample mean for n trials minus the mean of the parent random variable, which is fixed) is centered at a small interval around 0. More specifically, for an n large enough, there is a very low probability ($\delta$) that the probability distribution $\bar{X}_n - \mu$ takes values outside some ($\epsilon$) interval around 0. When I think of this, I visualize the normal distribution clustering around 0, where the area in the tails can be arbitrarily small. In this way, is it correct that the weak law does not guarantee that any given realization $\bar{x}_n-\mu$ of the random variable outside an epsilon interval of 0 is impossible...in fact there are potentially infinite such realizations (for n and for values greater than n) as one could visualize with the tails of a Gaussian?

I feel like this is where the strong law comes in, but I'm a little unclear on it. The statement of the strong law is $Pr(\lim_{n \to {\infty}}\bar{X}_n = \mu)=1$. I see that the random variable here is whether or not the sample mean converges to $\mu$ as n gets infinitely large (that is clusters around $\mu$), and the probability of such convergence is 1. So are we really looking at the distribution of sequences? How does this relate to, resolve, or expand upon the weak law of large numbers?

• Are you already aware that convergence almost surely implies convergence in probability, but not conversely? And do you know the counterexamples showing that the converse fails? If not, they may be illuminating. – grndl Sep 15 '16 at 23:46
• @aduh I've seen the proof and some counterexamples. I see that almost surely convergence is talking about point wise convergence, where the set of input values that you are measuring are possible outcomes (e.g a string of heads/tails). But I'm having trouble visualizing a.s convergence as I did weak convergence in the sense of the example of diminishing tails on a Gaussian-which I suppose may not be apt. – Winston Sep 16 '16 at 13:31
• Do you have a preferred way of visualizing pointwise convergence? The way I like to think about a.s. convergence is to just pretend that it is pointwise, keeping in the back of my mind that there are some points that fail to converge but they're "negligible" (measure 0). – grndl Sep 16 '16 at 14:48
• @aduh I agree, that is a nice way to think about it. But I suppose what I wanted to find is a way to think of the a.s convergence with a more math stats application. The weak law can as I understand it be thought of as the probability distribution of a random variable, the sample mean, being clustered around the true mean parameter, with a low probability of deviating too far from it. Can we say something about a probability distribution for say teaching purposes that can illuminate the meaning of the strong law and how it "improves" upon the weak law? – Winston Sep 16 '16 at 16:32
• @Winston, I have the same question...How to visualize SLLN? Many recourses online simply use Borel-Cantelli lemma to prove SLLN, and arguing about the counter example where convergence in probability doesn't imply convergence a.s. I still fail to have a picture as to how convergence a.s. works. I know that convergence a.s. is like point wise convergence, but what are the points here?What is the underlying probability space? – Sean Ian Nov 20 '18 at 18:03