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Variance adds for every finite pairwise-independent family

Statement

If $(X_{i})_{i \in I}$ is a finite pairwise-independent family of square-integrable real random variables on a common probability space, then $Var ⁣ (\sum_{i \in I} X_{i}) = \sum_{i \in I} Var (X_{i}) .$ Pairwise independence, rather than mutual independence, is sufficient. In particular, the original finite-probability-space case is included, with pairwise atom independence as in the finite-valued convention. The empty sum has variance zero.

Facts & Assumptions

Given: A finite pairwise-independent family $(X_{i})_{i \in I}$ of square-integrable real random variables on one probability space.

[L1]

Expectations of products of independent integrable real random variables factor; the finite-valued atom-independence convention is included. (Expectation factors over a finite product of mutually independent random variables)

[L2]

For square-integrable real random variables, covariance equals $E [X Y] - E [X] E [Y]$ , is symmetric and bilinear on finite linear combinations, and variance equals the self-covariance. These identities agree with the original finite-space definitions. (Variance and covariance identities for random variables)

Proof

technique · direct

1.1givenL1L2algebra

Each $X_{i}$ is integrable because $2 ∣ X_{i} ∣ \leq 1 + X_{i}^{2}$ and the measure has total mass one. For distinct $i, j$ , pairwise independence and [L1] give $E [X_{i} X_{j}] = E [X_{i}] E [X_{j}]$ , so [L2] gives $Cov (X_{i}, X_{j}) = 0$ .

2.1step 1.1L2algebra∎

The finite sum $S = \sum_{i \in I} X_{i}$ is square-integrable: repeated use of $(a + b)^{2} \leq 2 a^{2} + 2 b^{2}$ gives an integrable bound for $S^{2}$ . By [L2], $Var (S) = Cov (S, S) = \sum_{i, j \in I} Cov (X_{i}, X_{j}) = \sum_{i \in I} Var (X_{i}),$ where step 1.1 removes all off-diagonal terms. The empty sum is the constant zero, and the singleton case is the identity.

Depends on

Used by

Dependency tree · two levels

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Sources

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 6.4 (standard reference, not scraped)
J. Matousek and J. Vondrak, The Probabilistic Method, Section 6.1 (standard reference, not scraped)