I am am trying to understand this proof of why we can't deduce the law of large numbers from the Central limit theorem.
The central limit theorem states that the mean of i.i.d. variables, as N goes to infinity, becomes normally distributed.
I know I will be wrong but that makes me think that for i.i.d. random variables $X_i$ with mean $\mu$ and standard deviation $\sigma$, the cumulative distribution function $F_{Z_n}(a)$ of $$Z_n = \frac{1}{n} \sum_{i=1}^n X_i$$ converges to the cumulative distribution function of $\mathcal N(\mu,\sigma)$, a normal random variable with mean $\mu$ and standard deviation $\sigma$. Or, the distribution of $Z_n - \mu$ converges to the distribution of $\mathcal N(0,\sigma)$, or the distribution of $(Z_n - \mu)/\sigma$ converges to the distribution of $\mathcal N(0,1)$, the standard normal random variable.
I don't understand why this statement implies that:
$$P\{|Z_n - \mu| > \sigma\} = 1 - F_{Z_n}(\mu + \sigma) + F_{Z_n}((\mu + \sigma)^-) \to 1-\Phi(1)+\Phi(-1) \approx 0.32$$ as $n \to \infty$.
In particular I don't understand:
- why the probability of the sandardized normal distribution of n variables minus its to be greater than its variance is eaqual to 1 - the cumulative distribution of thea mean minus plus the variance plus the cumulative distribution of its opposite. That is to say: $$P\{|Z_n - \mu| > \sigma\} = 1 - F_{Z_n}(\mu + \sigma) + F_{Z_n}((\mu + \sigma)^-)$$
- And I don't undersand this notation $F_{Z_n}((\mu + \sigma)^\bbox[5px,border:3px solid red]{-})$. Does it means it goes to $-(\mu + \sigma)$?
I am a data scientist but I come from literature background. I would be very happy if you could explain it to me very simply, like to a teenager, or even with explicit examples or charts.