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 |
$\blacksquare$