# 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

- 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel:
*Time Series Analysis: Forecasting and Control*(3rd ed.) ... (previous) ... (next):

- 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

- $2.1$ Autocorrelation Properties of Stationary Models:

- $2$: Autocorrelation Function and Spectrum of Stationary Processes:

- Part $\text {I}$: Stochastic Models and their Forecasting: