I have heard “independent identically distributed (iid) process” a lot. It means mathematically that, for any $t$ and $t’$, $X(t)$ and $X(t’)$ follows the same distribution and independent each other. Then, it can be also considered an ergodic process.
As an example in my field (i.e., from the communication engineer perspective), there is a Rayleigh wireless channel, which is modeled as an iid process. In detail, for all $t$, $X(t) \sim \mathcal{CN}(0,1)$.
However, I have never heard yet the nonindepemdent identically distributed (niid) process, and finally faced it in a paper now. Although its “literal” meaning can be understood, I cannot understand it well.
Can someone explain its physical meaning? Or can someone give me some examples?
Non-independent identically distributed is what one would typically expect if $X(t)$ is a stationary Markov process, or the output of a stable filter driven by an i.i.d. input signal.
For example, suppose $Y(t)\sim\mathcal{CN}(0,1)$ is an iid "noise" sequence and the signal of interest is $$X(t)=\sum_n a_n Y(t-n),$$ where $(a_n)$ is some fixed, non-random, sequence of complex numbers numbers such that $\sum_n |a_n|^2=1$. The $X(t)$ sequence will typically be not iid. But if the $(a_n)$ sequence is such that all $a_n=0$ except for a single particular value of $n$, then the $X(t)$ sequence will be iid, after all.