Wide Sense Stationary Process

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Wide Sense Stationary Process

Definition (Wide Sense Stationary Process)

A sequence of RVs $X_{1},\dots,X_{n},\dots$ with finite variance is wide sense stationary (WSS) if

Constant Mean: $E[X_{k}]=\mu$ $\forall k$ (we assume $\mu=0$ )
Stationary Autocorrelation: $E[X_{k}X_{j}]=E[X_{k+i}X_{j+i}], \ \ \forall i,j,k$ i.e. this value only depends on the difference between indices.

Remark

Define $r_{j}=E[X_{n}X_{n-j}]$ and note that $r_{j}=r_{-j}$ . This is the autocorrelation function of our process. We assume $r_{0}=E[X_{n}^{2}]>0$ since otherwise $X_{n}=0$ w.p.1. (this is due to stationarity).

Linked from

Predictive Quantization vs Scalar Quantization

WSS Predictor Coefficients