Zero-mean time series models

IID Noise

A Discrete $\{X_i:i=1,2\dots n\}$ with:

• Independence property is satisfied: joint distribution = product of marginals.
• $E[X_i]=0\forall i$

White Noise

Sequence of uncorrelated random variables $\{X_1,X_2,\dots \}$ with zero mean and finite variance. Denoted $X_i\sim WN(0,\sigma )$

• $E(X_i)=0$
• $Cov(X_i,X_j)=0$ $\forall i\ne j$

Connection

What is the relationship between white noise and iid noise?

Independence is a stronger condition than non-correlation. For independence we need to know joint distributions but for non-correlation we don’t. So we’d rather work with white noise in Forecasting.