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Given $\Cov(a X_1 + bX_2, cY_1 + d_Y2) = ac \Cov(X_1, X_2) + ad \Cov(X_1, Y_2) + bc \Cov(X_2, Y_1) + bd \Cov(X_2, Y_2)$ or more general \(\Cov(\sum_{j=1}^{s} a_j W_j, \sum_{k=1}^{t} b_k W_k) = \sum_{j=1}^{s} \sum_{k=1}^{t} a_j b_k \Cov(W_j, W_k)\)
Autocovariance is defined as
\[\gamma_X(s, t) = \Cov(X_s, X_t) = \E[(X_s - \E[X_s])(X_t - \E[X_t])]\]Given weakly stationary time series, then for all $t \in T$, $\E[X_t] = \mu$ and $\Var[X_t] = \sigma^2 \abs{t-s}$
Given two time series ${X_s}$ and ${Y_t}$ with $\Var(X_s) < \infty$ and $\Var(Y_t) < \infty$, the autocorrelation is defined as
\[\gamma_{X, Y}(s, t) = \Cov[(X_s - \E[X_s])(Y_t - \E[Y_t])]\]