I'm trying to find concrete examples of the SLLN theorem. Before, let's see the statement of this theorem precisely from this book, page 81:
Definition: We say that $X_n$ converges almost surely to $X$, written $X_n\xrightarrow{a.s.} X$, if
$$\mathbb{P}(\{s:X_n(s)\to X(s)\})=1$$
Theorem (The Strong Law of Large Numbers). Let $X_1,\ldots X_n$ be IID. If $\mu=\mathbb E|X_1|\lt \infty$ then
$$\bar X_n\xrightarrow{a.s.}\mu$$
I'm thinking about for example the $X\sim Bern(1/2)$, such as the coin experiment where we can define as $X(T)=0$ and $X(H)=1$.
The way I understand the theorem is it states for every $s$ in the outcome from the experiment (except the ones with $P(s)=0$), we have $\bar X_n(s)\to \mu$.
I can't see this in my example. Let's start with $s=H$. Since we already know $\mu =1/2$, we have:
$$\bar X_n(H)=\frac{X_1(H)+\ldots+X_n(H)}{n}=\frac{1+\ldots+1}{n}=1\nrightarrow 1/2$$
as $n$ goes to infinity.
Where am I wrong?
$\endgroup$ 101 Answer
$\begingroup$Given that the proof of SLLN is well known and you can find it in a lot of books (Here, for example, you can find a very basic proof) I would like to make you think about an intuitive reasoning:
SLLN in your example can be stated in the following way: For $n$ large, calculate the following probability
$$\mathbb{P}\Bigg[\lim\limits_{n}\overline{X}_n=\frac{1}{2}\Bigg]=?$$
Say: toss the fair coin $10,000,000$ times: what is the probability you observe a numbers of H $\rightarrow 5,000,000$?
...I think it is intuitive that this probability is 1.
SLLN provides a proof of this intuition
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