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Characteristic functions under affine maps and independent sums
Statement
For real , . For a finite mutually independent family of real random variables, . The empty sum has characteristic function one.
Facts & Assumptions
Given: The hypotheses and conventions in the statement.
The characteristic function is the expectation of the exponential. Characteristic function of a real random variable.
The exponential of a sum is a product. , and the complex exponential extends the real exponential.
Real integrable Borel coordinate functions factor over independent variables. Expectations factor over finite products of independent random variables.
Finite complex linear combinations commute with integration. The Lebesgue integral is linear on .
Proof
The addition law gives . Both random exponentials are bounded and integrable. Pulling out the constant gives the affine identity, including and .
For put and . Expand 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
For the sum is zero, its exponential is one, and the empty product is one. For the asserted identity is the defining expectation itself.
Depends on
Used by
- Deterministic Ito integrals are Gaussian Corollary
- One-sample natural error in the Gaussian model Corollary
- A boundary null has a nonstandard likelihood-ratio limit Counterexample
- Infinite variance can defeat square-root-n CLT scaling Counterexample
- Pointwise resampling misses the long-run variance of an AR(1) mean Counterexample
- Two visits can alias random-slope and residual variance components Counterexample
- d-dimensional Brownian motion Definition
- Multivariate normal law, including singular covariance Definition
- A degenerate multivariate Gaussian limit Example
- A hidden common cause is observationally indistinguishable from an edge Example
- A stationary Ornstein–Uhlenbeck diffusion reverses to itself Example
- At the critical quadratic ULA step size, no invariant law exists Example
- Characteristic function of a gaussian law Example
- Characteristic functions of bernoulli binomial and poisson laws Example
- Gaussian shift LAN and the regular sample mean Example
- Independent sums via characteristic functions Example
- Known-variance normal mean and exact likelihood calibration Example
- PC distinguishes a chain from an unshielded collider Example
- Stationary Ornstein–Uhlenbeck score and probability flow Example
- A finite stack of independent Gaussian vectors is jointly Gaussian Lemma
- A nonconstant centered integrable IID random walk is unbounded above Lemma
- A shared column of an independent orthogonal mixing must be Gaussian Lemma
- Brownian covariance is equivalent to independent stationary normal increments Lemma
- Characteristic function of a normal law Lemma
- Conditional CLT for independent resampled weighted blocks Lemma
- Endpoint law and restriction density of a scalar Brownian bridge Lemma
- Invariant Gaussian law and uniqueness for a stable autoregression Lemma
- Signal alignment of a Gaussian sample mean Lemma
- The mean of a finite independent normal sample is normal Lemma
- The pCN Gaussian reference pair law is symmetric Lemma
- A positive-definite multivariate normal law has its Gaussian density Theorem
- Exact increment laws in the dyadic Brownian bridge tree Theorem
- Feller converse to Lindeberg-Feller Theorem
- Independent-group terminal influence limit Theorem
- Lindeberg-Feller central limit theorem: sufficiency Theorem
- Lindeberg-Levy iid central limit theorem Theorem
- Projection CLT for a two-sample U-statistic Theorem
- Robust sample lower bound in the Gaussian model Theorem
- Robust sample upper bound in the Gaussian model Theorem
- The finite linear-Gaussian state-space path is jointly Gaussian Theorem
…and 2 more results.
Dependency tree · two levels
26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)
- Norris, Probability and Measure (standard reference, not scraped)