The white noise model is sequence of [[Independence and Identical Distribution|i.i.d.]] [[Random Variable|Random Variables]] $W_t$, where each of these terms exhibits zero mean, the same [[Variance]] and zero [[Correlation]] among each other $(\forall \: t)$. $ \begin{align} \mu(t) &= \mathbb E[W_t] =0 &&\forall t \\[6pt] \gamma_W(t,t) &= \mathrm{var}(X_t)=\sigma_W^2 && \forall t\\[6pt] \gamma_W(s,t) &= \mathrm{cov}(W_s, W_t) =0&&\forall s \not = t \end{align} $ Thus a time series $X_t$ of such a model writes as follows: $ X_t = W_t, \quad W_t \stackrel{\text{iid}}{\sim}(0, \sigma_W^2) $