> [!tldr] > A random walk $(X_{t})$ is a discrete random process where the "step" $Z_{t}:= X_{t}-X_{t-1}$ is a [[Purely Random Processes|purely random process]] (normality of $Z_{t}$ is not assumed). > > Equivalently, a random walk $(X_{t})$ has a purely random first difference: $\nabla X_{t}=Z_{t}$. If the PRP $(Z_{t})$ has mean $\mu$ and variance $\sigma^{2}_{Z}$, then $\mathbb{E}[X_{t}]=\mu t$ and $\mathrm{Var}(X_{t})=t\sigma_{Z}^{2}$. ```R fold dev.off() prp <- rnorm(500, mean=0, sd=1) data <- cumsum(prp) par(mfrow=c(2,1), mar=c(3,4,3,4), bg=NA) plot(data, type="l", ylab="Series Value", main="Random Walk") acf(data, xlab="Time", ylab="Autocorr.", main="") ``` ![[RandomWalkACF.png#invert]]