I am studying a tutorial on stochastic processes and there's an example in it which I don't understand anything of it.
First of all there is this criterion for a mean-ergodic random process:
For a WSS random process to be ergodic in the mean, the variance of the sample mean $$\operatorname{var} (\hat\mu_N)=\frac{1}{N}\sum_{k=-(N-1)}^{N-1}\left(1-\frac{|k|}{N}\right)(r_X[k]-\mu^2)$$ must converge to zero as $N\to\infty$.
1- What is $r_X[k]$ in the formula and how is it computed?
Then there is the following example:
Define a random process as $X[n]=A$ where $A=N(0,1)$. The random process is WSS since $$\mu_X[n]=E[X[n]]=E[A]=0=\mu$$ $$r_X[k]=E[X[n]X[n+k]]=E[A^2]=1$$ However, it should be clear that sample mean will not converge to $\mu$
In addition, it can be shown that var(sample mean)=1
Each realization is not representative of the ensemble of realizations.
Assuminig that $A=N(0,1)$ is the standard normal distribution
2- How is it clear that sample mean does not converge to $\mu$?
3- Why var(sample mean)=1?
$r_X[k]$ is auto-correlation of WSS random process $X[k]$ given by $$r_X[k]=\frac{E[(X[t]-\mu)(X[t+k]-\mu)]}{\sigma^2}$$ where $\mu=E[X[t]]$ and $\sigma^2=D(X[t])$ are expectation and variance of random process $X$.
In this case, sample mean $\hat{\mu}_N=A$ for all $N$ and the sequence $A,A,\ldots$ almost surely does not converge to $\mu$ (it converges to $\mu$ only when $A=\mu$ but $P\{A=\mu\}=0$).
Variance $\mathrm{Var}[\hat{\mu}_N]=\mathrm{Var}[A]=1$ as $A$ is standard normal random variable.