According to the definition of (strict) stationary random process "A process is strictly stationary if the joint distribution of any set of random variables obtained by observing the random process $X(t)$ is invariant wrt location of origin t = 0". A corollary of this definition is $F_{X(t)} (x) = F_{X(t+\tau ) } (x)$ for all $\tau$ and t. If I keep on making $\tau$ very small, am I right in inferring that $F_{X(t_1 ) }(x) = F_{X(t_2 )} (x)$. If yes then the auto correlation should always be constant , that is equal to $R_X (0)$ Also please provide (if possible) a 3-D plot of stationary process.


Your claimed corollary

$F_{X(t)} (x) = F_{X(t+\tau ) } (x)$ for all $t$ and $\tau$

is being interpreted incorrectly. It is true that all the random variables in the process have the distribution function, but in order to calculate the covariance of two of these variables, you need their joint distribution, and, for strictly stationary random processes, this joint distribution depends only on $\tau$, the separation in time of the two random variables. Nowhere is it claimed that (for fixed $t$), the joint distribution of $X(t)$ and $X(t+\tau)$ is the same for all choices of $\tau$. Indeed, the invariance to time origin for strictly stationary random processes is saying that

For all choices of real number $T$, the joint distribution of $X(t$) and $X(t+\tau)$ --- two random variables separated in time by $\tau$ --- is the same as the joint distribution of $X(t+T$) and $X(t+\tau+T)$ which also are two random variables separated in time by $T$.

It is not saying that $X(t)$ and $X(t+\tau)$ have the same joint distribution for all choices of real number $\tau$.

See this answer of mine on our sister site dsp.SE for more information about the consequences of strict stationarity.

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