Why doesn't law of large numbers apply to this stationary time-series?

1. Which situation uses the law of large numbers?
2. Why does the law of large numbers work?
3. Why is it important for time series to be stationary?

Which situation uses the law of large numbers?

For example, a fair coin toss is a Bernoulli trial. When a fair coin is flipped once, the theoretical probability that the outcome will be heads is equal to 1⁄2. Therefore, according to the law of large numbers, the proportion of heads in a "large" number of coin flips "should be" roughly 1⁄2.

Why does the law of large numbers work?

The law of large numbers, in probability and statistics, states that as a sample size grows, its mean gets closer to the average of the whole population. This is due to the sample being more representative of the population as the sample become larger.

Why is it important for time series to be stationary?

Stationarity is an important concept in the field of time series analysis with tremendous influence on how the data is perceived and predicted. When forecasting or predicting the future, most time series models assume that each point is independent of one another.