characteristic function of a probability distribution

特征函数

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Translation evidence: Literature usage

Meaning in context

For a real random variable XX, the function φX(t)=E[eitX]\varphi_X(t)=E[e^{itX}] for t∈Rt\in\mathbb R, using this sign convention.

实随机变量的复指数exp(itX)的期望所组成的函数，t取任意实数。

Example

For a fair sign XX, φX(t)=cos⁡t\varphi_X(t)=\cos t.

Usage and distinctions

This is not an event indicator, which analysis texts may also call a characteristic function; it exists without moment assumptions.

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Definition sources

  1. Gregory F. Lawler — Notes on ProbabilitySection 5, Central limit theorem, characteristic-function definition · Checked 2026-10-14

Translation sources

  1. 李东风 — 应用随机过程（北京大学）§1.3.5 特征函数，定义1.13 · Checked 2026-10-14
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