Variance of Linear Function of Observations of Stationary Process

Theorem
Let $S$ be a strictly stationary stochastic process giving rise to a time series $T$.

Let $\sequence {s_n}$ be a sequence of $n$ successive values of $T$:
 * $\sequence {s_n} = \tuple {z_1, z_2, \dotsb, z_n}$

Let $L_t$ be a linear function of $\sequence {s_n}$:
 * $L_t = l_1 z_t + l_2 z_{t - 1} + \dotsb + l_n z_{t - n + 1}$

Then the variance of $L_t$ is given by:


 * $\var {L_t} = \ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n l_i l_j \gamma {\size {j - i} }$

where $\gamma_k$ is the autocovariance of $S$ at lag $k$.