Why doesn't law of large numbers apply to this stationary time-series? There's a paragraph in Wikipedia that states the following:


  Let Y be any scalar random variable, and define a time-series $\{X_t\}$, by
  $$X_{t}=Y\qquad {\text{ for all }}t$$
  Then $\{X_t\}$ is a stationary time series, for which realisations consist of a series of constant values, with a different constant value for each realisation. A law of large numbers does not apply on this case, as the limiting value of an average from a single realisation takes the random value determined by $Y$, rather than taking the expected value of $Y$.


I cannot understand the reason for the law of large numbers not being applicable. Any explanation would help.
I believe that you are thinking that each value of $X_t$ is determined by a different realisation of $Y$, which in this example is not true.

Suppose that $Y$ is the value that comes out from a dice throw. Thus, $Y$ can achieve any integer value between $1$ and $6$. Throw the dice once. Suppose that you get $2$. Then, according to the definition in the example, your time-series would be:

$$X_t=2 \qquad {\text{for all }}t$$

What I think the quote from Wikipedia is saying is that the mean value of this stochastic process doesn't coincide with the mean value of the random variable $Y$, as

$$\mathbb{E}[X_t]=2 \neq 3.5 = \mathbb{E}[Y]$$

If you constructed the time-series letting each value of $X_t$ be determined by a new throw of the dice, then the means would match.