I came across a question in which the following figure was given and the question was: "The autocorrelation function RXX[k1,k2] of a zero-mean random process X[k] is depicted below. Which of the following properties are fulfilled?". The answer is: it's a wide-sense cyclostationary with period 2.
How can we conclude this based on the autocorrelation function? In general, if the autocorrelation matrix is diagonal, what can we conclude about the underlying process?
By inspection, you can validate that the autocorrelation function is given by the formula:
$$R_{xx}[k_1,k_2] = \sum_{q = -\infty}^{\infty} A \cdot \delta[k_1-2q]\delta[k_2-2q]$$
where A is the amplitude of each unit impulse you see in the figure. $$\delta[n] =\left\{\begin{aligned} &1, n=0\\ &0, n\in \mathbb{Z}-\{0\} \end{aligned} \right.$$ To elaborate on that, notice that only when: $k_1 = k_2 = 2q$ for some $q \in \mathbb{Z}$, will $R_{xx}$ be non zero.
To show that the process is Wide Sense Cyclostationary 2 conditions should be proven:
1) For $T = 2$ and $\forall m \in \mathbb{Z}$: $$\begin{aligned} R_{xx}[k_1+mT,k_2+mT] &= \sum_{q = -\infty}^{\infty} A \cdot \delta[k_1+mT-2q]\delta[k_2+mT-2q]\\ &= \sum_{q = -\infty}^{\infty} A \cdot \delta[k_1+2m-2q]\delta[k_2+2m-2q]\\ & = \sum_{q-m =-m -\infty}^{-m+\infty} A \cdot \delta[k_1-2(q-m)]\delta[k_2-2(q-m)]\\ &= \sum_{p = -\infty}^{\infty} A \cdot \delta[k_1-2p]\delta[k_2-2p]\\ &= R_{xx}[k_1,k_2] \end{aligned}$$
2) Now by the assumption made we know: $$\mathbb{E}[\textbf{x}[k]]= 0 \text{, }\forall k \in \mathbb{Z}$$ Therefore for $T = 2$ and $\forall m\in \mathbb{Z}$ $$\mathbb{E}[\textbf{x}[k]] = \mathbb{E}[\textbf{x}[k+mT]]$$
To conclude this process obeys the definition of wide sense cyclostationarity.
Regarding the last question on generalizing the results for a "diagonal" autocorrelation function, we would have a process with: $$R[k_1, k_2] = \sum_{q = -\infty}^{\infty}A_q \delta[k_1-q]\delta[k_2-q]$$ I cannot say something more without making further assumptions because the amplitudes $A_q$ may vary in general.