Consider the following simple model for Seattle weather forecast. The
weather can be either ‘rain’ or ‘no rain’. Suppose that the chance of rain tomorrow
only depends on whether it rains today or not, and not on past weather conditions.
If it rains today, the chance it will be rain tomorrow again is α ∈ (0, 1); and if it
does not rain today, the probability it will rain tomorrow is β ∈ (0, 1). Starting
from some day zero, let X0, X1, X2, . . . be a sequence where
Xk =
{1, if it rains on day k
0, otherwise