> [!tldr] > A discrete purely random process is a sequence of iid. distributed variables $(Z_{t})$. They are usually assumed to be normally distributed with mean $\mu=0$ and variance $\sigma^{2}_{Z}$. ### Properties of PRP's If assumed to follow $N(0,\sigma^{2}_{Z})$, a PRP is a (strictly) stationary normal process. Per definition, the purely random process $(Z_{t})$ has a constant [[Autocorrelations|acv.f.]] of $\gamma(\tau):=\mathrm{Cov}(Z_{t},Z_{t+\tau})=\begin{cases} \sigma^{2}_{Z} & \text{if }k=0 \\ 0 & \text{otherwise} \end{cases}$and its [[Autocorrelations#Correlograms|correlogram]] has no strong correlations for any lag $\tau>0$: ```R fold data <- rnorm(500, mean=0, sd=1) par(mfrow=c(2,1), mar=c(3,4,3,4), bg=NA) plot(data, type="l", ylab="Series Value", main="Purely Random Normal Process") acf(data, xlab="Time", ylab="Autocorr.", main="") ``` ![[PRPNormalACF.png#invert]]