# sufficient conditions for a stochastic process to be wide sense stationary

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?

• You should include what you think are the definitions of the autocorrelation function and autocovariance function of a random process. Your "familiar form" $$R(\tau)=\frac{E[(X_t-\mu)(X_{t+\tau}-\mu)]}{\sigma^2}$$ would be called the normalized autocovariance function by those who define the autocorrelation function as $R(t_1, t_2) = E[X_{t_1}X_{t_2}$ and for wide-sense-stationary processes, $$R_X(\tau) = R(t,t+\tau) = R(0,\tau).$$ Aug 3, 2015 at 19:13

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.
• Is my conclusion true? we have $WSS\Rightarrow \frac{E[(X_t-\mu)(X_{t+\tau}-\mu)]}{\sigma^2}=f(\tau)$ so being $WSS$ is the necessary condition for $\frac{E[(X_t-\mu)(X_{t+\tau}-\mu)]}{\sigma^2}=f(\tau)$ and $\frac{E[(X_t-\mu)(X_{t+\tau}-\mu)]}{\sigma^2}=f(\tau)$ is the sufficient condition for being $WSS$. So in order to prove that a stochastic process is $WSS$ it is enough to show that $\frac{E[(X_t-\mu)(X_{t+\tau}-\mu)]}{\sigma^2}=f(\tau)$ Aug 3, 2015 at 20:15
• No, because $\mu$ and $\sigma^2$ are well-defined only if you've proven that they are time-independant. So you have to prove the two points I gave in my answer. Tell me what's in contradiction with what I said in your tutorial. Aug 3, 2015 at 20:22