> [!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]]