# Covariance of sums of time series random variables [duplicate]

I find the following in my Cryer and Chan Time Series Analysis with application in R textbook:

Let $$c_{1}, c_{2},\ldots, c_{m}$$ and $$d_{1}, d_{2},\ldots, d_{n}$$ be constants and $$t_{1}, t_{2},\ldots, t_{m}$$ and $$s_{1}, s_{2},\ldots, s_{n}$$ be time points. Further let $$Y$$ be a random variable indexed with the previously mentioned time points.

Then:

$$Cov\:\left(\sum_{i = 1}^{m}c_{i}Y_{t_{i}} \:,\:\sum_{j = 1}^{n}d_{j}Y_{s_{j}}\right) = \sum_{i = 1}^{m} \sum_{j = 1}^{n}c_{i} d_{j}\:Cov(Y_{t_{i}} \: Y_{s_{j}})$$

How is this be proven? The book says that it follows from the linear properties of expectation, but I'm not sure how this was used here.

Further, the book states:

$$Var\:\left(\sum_{i=1}^{m}c_{i}Y_{t_{i}}\right) = \sum_{i = 1}^{m}c_{i}^{2}\:Var(Y_{t_{i}}) + 2 \sum_{i = 2}^{m} \sum_{j = 1}^{i - 1}c_{i}c_{j}\:Cov(Y_{t_{i}} , Y_{t_{j}})$$

Which is a special case of the first result. Why is this a special case of the first result?

• What's the covariance of a variable with itself? Dec 27, 2021 at 14:12
• This is the multilinearity property of covariance.
– whuber
Dec 27, 2021 at 18:48
• The second part of this question is dealt with in the answers to this question from some time ago, and I am sure that the first question has also been answered in detail somewhere on stats.SE. Dec 27, 2021 at 19:21

By definition, $$Cov(Y_{t_i},Y_{s_j}) = E(Y_{t_i}Y_{s_j}) - E(Y_{t_i})E(Y_{s_j})$$. Applying this definition to the first equation gives $$\begin{eqnarray*} Cov \left(\sum_{i=1}^m c_i Y_{t_i},\sum_{j=1}^n d_j Y_{s_j}\right) &=& E \left(\sum_{i=1}^m c_i Y_{t_i} \sum_{j=1}^n d_j Y_{s_j}\right) - E\left(\sum_{i=1}^m c_i Y_{t_i}\right) E\left( \sum_{j=1}^n d_j Y_{s_j}\right) \\ &=& \sum_{i=1}^m \sum_{j=1}^n c_i d_j E\left(Y_{t_i}Y_{s_j}\right) - \sum_{i=1}^m \sum_{j=1}^n c_i d_j E\left(Y_{t_i}\right)E\left(Y_{s_j}\right) \\ &=& \sum_{i=1}^m\sum_{j=1}^n c_i d_j Cov(Y_{t_i},Y_{s_j}). \end{eqnarray*}$$
The second equation you give is a special case of the above since $$\begin{eqnarray*} Var \left(\sum_{i=1}^m c_i Y_{t_i}\right) &=& Cov\left(\sum_{i=1}^m c_i Y_{t_i},\sum_{i=1}^m c_i Y_{t_i}\right). \end{eqnarray*}$$ Use the previous formula to simplify the above and separate the double summation into two cases $$(i=j)$$ and $$(i \ne j)$$.