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.