Covariance of Sums of Random Variables

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Theorem

Let $n$ be a strictly positive integer.

Let $\sequence {X_i}_{1 \mathop \le i \mathop \le n}$, $\sequence {Y_j}_{1 \mathop \le j \mathop \le n}$ be sequences of random variables.


Then:

$\ds \cov {\sum_{i \mathop = 1}^n X_i, \sum_{j \mathop = 1}^n Y_j} = \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \cov {X_i, Y_j}$


Proof

\(\ds \sum_{i \mathop = 1}^n \sum_{j \mathop = 1}^n \cov {X_i, Y_j}\) \(=\) \(\ds \sum_{i \mathop = 1}^n \paren {\sum_{j \mathop = 1}^n \cov {Y_j, X_i} }\) Covariance is Symmetric
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \cov {\sum_{j \mathop = 1}^n Y_j, X_i}\) Covariance of Sums of Random Variables: Lemma
\(\ds \) \(=\) \(\ds \sum_{i \mathop = 1}^n \cov {X_i, \sum_{j \mathop = 1}^n Y_j}\) Covariance is Symmetric
\(\ds \) \(=\) \(\ds \cov {\sum_{i \mathop = 1}^n X_i, \sum_{j \mathop = 1}^n Y_j}\) Covariance of Sums of Random Variables: Lemma

$\blacksquare$