crosscovariances

Cross-covariances refer to the linear relationship between pairs of random variables or stochastic processes, capturing how one quantity varies with another across time, space, or components. For random vectors X in R^p and Y in R^q with means μ_X and μ_Y, the cross-covariance is the p by q matrix Cov(X, Y) = E[(X − μ_X)(Y − μ_Y)^T]. When X = Y, this reduces to the usual covariance matrix Var(X). The cross-covariance matrix is related to the variances and covariances of the vector components and satisfies Cov(X, Y) = Cov(Y, X)^T and Cov(X, X) is symmetric positive semidefinite.

In the context of stochastic processes, the cross-covariance function is defined as γ_XY(h) = Cov(X_t, Y_{t+h}) for lag h. If the processes are wide-sense stationary, γ_XY(h) depends only on h. The cross-covariance is related to the cross-correlation by normalization: ρ_XY(h) = γ_XY(h) / sqrt(γ_XX(0) γ_YY(0)). In general γ_YX(h) = γ_XY(−h). If X and Y are independent, γ_XY(h) = 0 for all h.

Estimation and interpretation: For observed sequences x_t and y_t, the sample cross-covariance at lag h is γ̂_XY(h) = (1/(n − |h|)) ∑_{t=1}^{n−|h|} (x_t − x̄)(y_{t+h} − ȳ). For multivariate time series, the cross-covariance is a matrix-valued function across components. Cross-covariances underpin many analyses, including vector autoregression, cross-spectral analysis, and signal-processing lead-lag studies.