# Does a stationary process implies a normal distribution of the data?

My understanding is ‘no’, a stationary process does not imply a normal distribution of the data. However I haven’t found a clear indication in my library or online. I am interested in other resources comparing stationary and heteroscedastic processes to data distributions.

I understand a stationary process joint probability distribution does not change over time. But does it necessarily mean that the shape of the distribution is normal?

There was a discussion about this on dsp.SE some months ago. My answer there might help resolve some of the issues that the OP has.

Added in response to @Alexis’s complaint:

In part, what I said there is as follows:

• All the random variables in the process have identical CDFs: $$F_{X(t_1)}(x) = F_{X(t_2)}(x)$$ for all $$t_1, t_2$$.
• Any two random variables separated by some specified amount of time have the same joint CDF as any other pair of random variables separated by the same amount of time. For example, the random variables $$X(t_1)$$ and $$X(t_1 + \tau)$$ are separated by $$\tau$$ seconds, as are the random variables $$X(t_2)$$ and $$X(t_2 + \tau)$$, and thus $$F_{X(t_1), X(t_1 + \tau)}(x,y) = F_{X(t_2), X(t_2 + \tau)}(x,y)$$
• Any three random variables $$X(t_1)$$, $$X(t_1 + \tau_1)$$, $$X(t_1 + \tau_1 + \tau_2)$$ spaced $$\tau_1$$ and $$\tau_2$$ apart have the same joint CDF as $$X(t_2)$$, $$X(t_2 + \tau_1)$$, $$X(t_2 + \tau_1 + \tau_2)$$ which as also spaced $$\tau_1$$ and $$\tau_2$$ apart. Equivalently, the joint CDF of $$X(t_1), X(t_2), X(t_3)$$ is the same as the joint CDF of $$X(t_1+\tau), X(t_2+\tau), X(t_3+\tau)$$
• and similarly for all multidimensional CDFs.

Effectively, the probabilistic descriptions of the random process do not depend on what we choose to call the origin on the time axis: shifting all time instants $$t_1, t_2, \ldots, t_n$$ by some fixed amount $$\tau$$ to $$t_1 + \tau, t_2 + \tau, \ldots, t_n + \tau$$ gives the same probabilistic description of the random variables. This property is called strict-sense stationarity and a random process that enjoys this property is
called a strictly stationary random process or, more simply, a stationary random process. Be aware that in some of the statistics literature (especially the parts related to econometrics and time-series analysis), stationary processes are defined somewhat differently; in fact as what are described later in this answer as wide-sense stationary processes.

Note that strict stationarity by itself does not require any particular form of CDF. For example, it does not say that all the variables are Gaussian.

Later in the same answer, I say about wide-sense-stationary (or WSS) random processes a.k.a. weakly stationary stochastic (or WSS) processes….

Note that the definition says nothing about the CDFs of the random variables comprising the process; it is entirely a constraint on the first-order and second-order moments of the random variables.

Thus once again, there is nothing in the definition of stationary processes (in the time series folks’ meaning of the word) that requires the random variables to be normally distributed,