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Mathematics · Glossary

What is characteristic function?

Definition 23.1 University Mathematics — Year 3 · Chapter 23 — Characteristic Functions and the Central Limit Theorem

The characteristic function of a real random variable XX is

φX(ξ)=E[eiξX]=∫Reiξx  ⁣dPX(x)(ξ∈R)\varphi_X(\xi) = \E\bigl[\eu^{\iu\xi X}\bigr] = \int_\R \eu^{\iu\xi x}\,\dd\P_X(x) \qquad (\xi \in \R)

(the transfer theorem computes it from the law; for a density ff, φX(ξ)=f^(−ξ)\varphi_X(\xi) = \hat f(-\xi) in Chapter 14’s convention).

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