Characteristic function

The characteristic function¹ $𝜒 (𝑘)$ of a Real random variable $𝑋 \sim 𝑤$ is the Fourier transform of the Probability density function $𝑤 (𝑥)$ or equivalently the Expectation of the function $𝑒 - 𝑖 𝑘 𝑋$ ² #m/def/prob

𝜒 (𝑘) = ⟨ 𝑒 - 𝑖 𝑘 𝑋 ⟩ = F {𝑤} (𝑘) = \int \infty - \infty 𝑤 (𝑥) 𝑒 - 𝑖 𝑘 𝑥 𝑑 𝑥

which is a complex analogue to the moment-generating function. This describes the distribution of $𝑋$ completely — the density function may be obtained using the reverse Fourier transform:

𝑤 (𝑥) = F - 1 {𝜒} (𝑥) = 1 2 𝜋 \int \infty - \infty 𝜒 (𝑘) 𝑒 𝑖 𝑘 𝑥 𝑑 𝑘

Using the Taylor series expansion of $𝑒 - 𝑖 𝑘 𝑋$ one obtains a further representation of $𝜒 (𝑘)$ in terms of moments:

𝜒 (𝑘) = \infty \sum 𝑛 = 0 (- 𝑖 𝑘) 𝑛 𝑛! ⟨ 𝑋 𝑛 ⟩

#state/tidy | #lang/en | #SemBr

German charakteristische Funktion ↩
2006, Statistische Mechanik, p. 5 ↩

Characteristic function

Footnotes