Strictly Stationary Stochastic Process/Examples/Constant Mean Level

Example of Strictly Stationary Stochastic Process
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.

Consider the expectation of $S$:


 * $\mu = \expect {z_t} = \displaystyle \int_{-\infty}^\infty z \map p z \rd z$

where $\map p z$ is the (constant) probability mass function of $S$.

It is necessary that $T$ has a constant mean level, so that:
 * The sample mean over a set of $N$ successive values $\set {z_1, z_2, \dotsb, z_N}$

is the same as:
 * the sample mean over any other set of $N$ successive values $\set {z_{1 + k}, z_{2 + k}, \dotsb, z_{N + k} }$

for an arbitrary lag $k$.