White noise

A white noise is a sequence of uncorrelated random variables with the same mean and the same variance. A stationary stochastic process is called white noise if:

• $\mathbb{E}[e(t,s)]=\mu$ $\forall t$
• $Var[e(t,s)]=\mathbb{E}[(e(t,s)-\mu)^2]=\lambda^2$ $\forall t$
• $\gamma_e(\tau)=\mathbb{E}[(e(t,s)-\mu)(e(t-\tau,s)-\mu)]=0$ $\forall \tau\neq0$: this means that the values are completely uncorrelated for different time instants. This gives a white noise the caracteristic of having erratic realizations and unpredictability (fast sign changes around the mean)

We can then use a compact notation for the covariance function of a white noise: $$ \gamma_e(\tau)= \begin{cases} \lambda^2 &\text{if } \tau=0 \newline 0 &\text{if } \tau\neq0 \end{cases} $$

We will denote: $e(t,s) \sim WN(\mu,\lambda^2)$.