Variance of Linear Function of Observations of Stationary Process

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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:

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

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


Proof


Sources

Part $\text {I}$: Stochastic Models and their Forecasting:
$2$: Autocorrelation Function and Spectrum of Stationary Processes:
$2.1$ Autocorrelation Properties of Stationary Models:
$2.1.3$ Positive Definiteness and the Autocovariance Matrix