Chapter Summary

Chapter 13 Summary: Introduction to Stochastic Processes

Key Points

• 1.

  A stochastic process $\{X(t) : t \in \mathcal{T}\}$ is a family of random variables indexed by time. It is fully characterized by its finite-dimensional distributions (fdds), which must satisfy Kolmogorov's consistency conditions.

• 2.

  Strict-sense stationarity (SSS) requires all fdds to be time-shift invariant. Wide-sense stationarity (WSS) requires only constant mean and lag-dependent autocorrelation — a much weaker and more practical condition.

• 3.

  For Gaussian processes, WSS implies SSS (Lemma 43), because Gaussian distributions are fully determined by their first two moments.

• 4.

  The autocorrelation $r_{XX}(\tau)$ of a WSS process satisfies: $r_{XX}(0) =$ average power, $r_{XX}(\tau) = r_{XX}^*(-\tau)$ (Hermitian symmetry), $|r_{XX}(\tau)| \leq r_{XX}(0)$ (maximum at origin), and non-negative definiteness.

• 5.

  The cross-correlation $r_{XY}(\tau)$ measures the linear dependence between two jointly WSS processes. For an LTI system, $r_{YX}[k] = (h * r_{XX})[k]$.

• 6.

  Ergodicity ensures that time averages converge to ensemble averages. The sufficient condition for mean-ergodicity is $c_{XX}(\tau) \to 0$ as $|\tau| \to \infty$.

• 7.

  These second-order tools — autocorrelation, cross-correlation, and the WSS/ergodicity framework — are the foundation for the power spectral density (Chapter 14), LTI system analysis (Chapter 15), and mean-square calculus (Chapter 16).