Strictly Stationary Stochastic Process/Examples/Constant Mean Level

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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 $S$:

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

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 $k$.


Sources

Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes:
$2.1$ Autocorrelation Properties of Stationary Models:
$2.1.2$ Stationary Stochastic Processes: Mean and variance of a stationary process: $(2.1.1)$