This is from the Wikipedia page on Stationary Processes:

Let Y be any scalar random variable, and define a time-series { Xt }, by Xt = Y for all t.

Then { Xt } 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.

Just for reference, in probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

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    $\begingroup$ The LLN does not hold because, here, $\frac1n\sum\limits_{t=1}^nX_t\to Y$ when $n\to\infty$ and not $\frac1n\sum\limits_{t=1}^nX_t\to E(X_1)$ (unless $Y$ is almost surely constant). Note that $E(X_1)=E(Y)$. $\endgroup$ – Did Aug 30 '18 at 14:19
  • $\begingroup$ I see But, why does the first condition you mentioned hold? Please excuse my ignorance, as I'm new to learning these concepts. $\endgroup$ – LumosMaxima Aug 30 '18 at 14:22
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    $\begingroup$ "why does the first condition you mentioned hold? " Well, because $X_t=Y$ for every $t$ hence $\frac1n\sum\limits_{t=1}^nX_t=Y$ for every $n$, with full probability. $\endgroup$ – Did Aug 30 '18 at 14:23
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    $\begingroup$ This is a special case. The values are something like $X_t=2$ for every $t$. The only thing that's random is the constant value of the time series. We roll the die once, and it come up $2$ and that determines the entire series. When we roll the die again, we get a different series. $\endgroup$ – saulspatz Aug 30 '18 at 14:27
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    $\begingroup$ "how this case if different from the cases where the law of large numbers do hold" The setting of the LLN is when one repeats the experiment. In @saulspatz's example, one throws the die once. For the LLN to be relevant, one should throw the die again and again, and record the sequence of results. $\endgroup$ – Did Aug 30 '18 at 14:29

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