LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Characteristic functions under affine maps and independent sums

Statement

For real $a, b, t$ , $φ_{a X + b} (t) = e^{i t b} φ_{X} (a t)$ . For a finite mutually independent family $(X_{j})_{j = 1}^{n}$ of real random variables, $φ_{\sum_{j} X_{j}} (t) = \prod_{j} φ_{X_{j}} (t)$ . The empty sum has characteristic function one.

Facts & Assumptions

Given: The hypotheses and conventions in the statement.

[F1]

The characteristic function is the expectation of the exponential. Characteristic function of a real random variable.

[F2]

The exponential of a sum is a product. $\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential.

[F3]

Real integrable Borel coordinate functions factor over independent variables. Expectations factor over finite products of independent random variables.

[F4]

Finite complex linear combinations commute with integration. The Lebesgue integral is linear on $L^{1} (μ)$ .

Proof

technique · direct

1.1F1F2F4

The addition law gives $e^{i t (a X + b)} = e^{i t b} e^{i (a t) X}$ . Both random exponentials are bounded and integrable. Pulling out the constant $e^{i t b}$ gives the affine identity, including $a = 0$ and $b = 0$ .

1.2F1F2F3F4

For $n \geq 1$ put $c_{j} = \cos (t X_{j})$ and $s_{j} = \sin (t X_{j})$ . Expand $e^{i t \sum_{j} X_{j}} = \prod_{j} (c_{j} + i s_{j}) = \sum_{A \subseteq {1, \dots, n}} i^{∣ A ∣} \prod_{j \in A} s_{j} \prod_{j \notin A} c_{j} .$ Each real factor is a bounded Borel function of its own coordinate, so the real factorization theorem applies to each of these finitely many products. Complex linearity then gives $E e^{i t \sum_{j} X_{j}} = \sum_{A} i^{∣ A ∣} \prod_{j \in A} E s_{j} \prod_{j \notin A} E c_{j} = \prod_{j} (E c_{j} + i E s_{j}) = \prod_{j} φ_{X_{j}} (t) .$

2.1F1∎

For $n = 0$ the sum is zero, its exponential is one, and the empty product is one. For $n = 1$ the asserted identity is the defining expectation itself.

Characteristic functions under affine maps and independent sums

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources