[Math] sufficient conditions for a stochastic process to be wide sense stationary

correlationcovariancerandom variablesstatisticsstochastic-processes

From the page Stationary process, I have the following definition:

WSS random processes only require that 1st moment and autocovariance
do not vary with respect to time

and from the page Autocorrelation:

If $X_t$ is a wide-sense stationary process then the mean $\mu$ and the
variance $\sigma^2$ are time-independent, and further the autocorrelation
depends only on the lag between $t$ and $s$: the correlation depends only
on the time-distance between the pair of values but not on their
position in time. This further implies that the autocorrelation can be
expressed as a function of the time-lag, and that this would be an
even function of the lag $\tau = s − t$. This gives the more familiar form $$R(\tau)=\frac{E[(X_t-\mu)(X_{t+\tau}-\mu)]}{\sigma^2}$$and the fact that this is an even function can be stated as $$R(\tau)=R(-\tau)$$

Now the question is what should I do that $X(t) = B + A \sin(\omega_0 t + \Phi)$ is WSS?

  • I should prove that the mean and autocovariance do not depend on time.

Or

  • I should prove that the mean does not depend on time and autocorrelation only depends on the lag of time.

Furthermore we know that $X(t) = B + A \sin(\omega_0 t + \Phi)$ is a process in the form $f(A,B,\Phi;t)$ are the autocovariance and autocorrelation a $3\times 3$ matrix or a scalar number?

Best Answer

You should prove:

  1. that $E[X_t]$ is a number independant of $t$, say $\mu$.
  2. that the autocovariance $E[(X_t-\mu)(X_{t+h}-\mu)]$ is only a function of $h$ (the lag) and not $t$.

If $X_t$ is a one-dimensional stochastic process, then the autocovariance and autocorrelation functions are also one-dimensionnal. Otherwise, these are matrix with the same dimension as the process.

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