If the random variables which make up \(y_t\) are uncorrelated, have means 0 and variance \(\sigma^2\), then \(y_t\) is stationary with the autocovariance function of

\[ \gamma_w(s,t) = \text{cov}(w_s, w_t)= \begin{cases} \sigma^2_w & s=t \\ 0 & \text{otherwise} \end{cases} \]

This type of series is referred to as **white noise**. The designation *white* originates from the analogy with white light and indicates that all possible periodic oscillations are present with equal strength (Shumway and Stoffer 2011).

A particularly useful white noise series is *Gaussian white noise*, wherein the \(w_t\) are independent identically distributed (*iid*) normal random variables, with mean 0 and variance \(\sigma^2\)

\[w_t \sim \text{iid } N(0, \sigma^2_w) \]

We may easily generate a Gaussian white noise in R using the `rnorm()`

function.

```
set.seed <- 133 # set seed for reproducibility
w <- rnorm(n = 200, mean = 0, sd = 1)
plot.ts(w, main = "Gaussian white noise series")
```

Let us plot the correlogram using the `acf()`

function. This time however, for a more convenient visualization, we embed the `acf()`

function into the `autoplot()`

function provided by the `ggfortify`

package.

```
library(ggfortify)
autoplot(acf(w, plot = F)) +
ggtitle('Serial Correlation of Gaussian white noise')
```

As expect the autocorrelation for any lag from 1 to 25 is not different from zero, as indicated by the blue dashed line. At lag 0 the autocorrelation is 1.