LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Variance of a centered random weighted sum

Statement

Let $n \geq 1$ . Let $W_{1}, \dots, W_{n}$ be mutually independent real random variables with finite second moments, means $E [W_{i}] = 0$ , and variances $σ_{i}^{2}$ . Let $X = (X_{1}, \dots, X_{n})$ be a real random vector whose coordinates have finite second moments, and suppose the random elements $W$ and $X$ are independent. The coordinates of $X$ need not be mutually independent. For $S = \sum_{i = 1}^{n} W_{i} X_{i}$ , the variable $S$ is square-integrable, $E [S] = 0$ , and $Var (S) = \sum_{i = 1}^{n} σ_{i}^{2} E [X_{i}^{2}] .$

Facts & Assumptions

Given: Mutually independent centered square-integrable real random variables $W_{1}, \dots, W_{n}$ and a square-integrable random vector $X = (X_{1}, \dots, X_{n})$ independent of the vector $W = (W_{1}, \dots, W_{n})$ .

[F1]

Independence of random elements means independence of their generated sigma-algebras. (Independent random elements)

[F2]

The joint law of a finite family of independent random elements is the product of their marginal laws. (Independent random elements have product joint law)

[F3]

Measurable coordinatewise maps preserve independence. (Measurable coordinatewise functions preserve independence)

[F4]

Expectations of products of independent integrable random variables factor, and the product is integrable. (Expectations factor over finite products of independent random variables)

[F5]

Sums, products, and lattice operations of measurable real-valued functions used here are measurable. (Arithmetic and lattice operations preserve measurability whenever they are defined)

[F6]

If $U, V \in L^{2} (P)$ , then $E [∣ U V ∣] \leq (E [U^{2}])^{1 / 2} (E [V^{2}])^{1 / 2}$ . (Cauchy-Schwarz for random variables)

[F7]

Expectation is integration against the probability measure. (Expectation of a nonnegative or integrable random variable)

[F8]

For a square-integrable real random variable $Y$ , $Var (Y) = E [(Y - E [Y])^{2}]$ . (Moments, variance, and covariance on a probability space)

[F9]

Expectation is linear in scalar multiples and binary sums of integrable random variables; repeated use gives the finite-sum rule. (Linearity, monotonicity, and the modulus bound for expectation)

Proof

1.1givenF1F4F6F9

Each $W_{i}$ and $X_{i}$ is integrable by [F6] applied with the constant random variable $1$ . Independence of the two random elements gives independence of $W_{i}$ and $X_{i}$ by [F1], so [F4] yields $E [W_{i} X_{i}] = E [W_{i}] E [X_{i}] = 0$ . Linearity [F9] over the finite index set gives $E [S] = 0$ .

1.2givenF2F3F4F5F6

Fix $i \neq j$ . The coordinate-product maps $(w, x) \mapsto w_{i} w_{j}$ and $(w, x) \mapsto x_{i} x_{j}$ are measurable by [F5]. Since the joint law of $(W, X)$ is the product law by [F2], these maps, one applied to each independent factor, give independent random variables $A = W_{i} W_{j}$ and $B = X_{i} X_{j}$ by [F3]. Both are integrable by [F6]. Also $E [A] = E [W_{i}] E [W_{j}] = 0$ by [F4] and mutual independence of the weights. Hence [F4] gives $E [W_{i} W_{j} X_{i} X_{j}] = E [A B] = 0$ .

1.3givenF2F3F4

For each $i$ , $W_{i}^{2}$ and $X_{i}^{2}$ are independent nonnegative random variables by [F2] and [F3]. Applying the nonnegative case of [F4] gives $E [W_{i}^{2} X_{i}^{2}] = E [W_{i}^{2}] E [X_{i}^{2}] = σ_{i}^{2} E [X_{i}^{2}] < \infty$ .

2.1step 1.1step 1.2step 1.3F7F8F9algebra∎

Expanding the finite square $S^{2}$ gives diagonal terms from step 1.3 and cross terms from step 1.2, so $S^{2}$ is integrable. Linearity [F9] gives $E [S^{2}] = \sum_{i} σ_{i}^{2} E [X_{i}^{2}]$ . By step 1.1, $E [S] = 0$ ; [F7] and [F8] therefore identify this second moment with $Var (S)$ and prove the claim.

Depends on

Used by

PReLU fan-in and fan-out variance scales Corollary
ReLU backward gradient-variance recurrence Theorem
ReLU forward preactivation-variance recurrence Theorem

Dependency tree · two levels

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Sources

Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun, Delving Deep into Rectifiers, §2.2 (standard reference, not scraped)
Simon J. D. Prince, Understanding Deep Learning, §7.5.1 (standard reference, not scraped)
Johns Hopkins University, Handout #8: Neural Network Training and Tokenization, §3 (standard reference, not scraped)