Characteristic function

A function that encodes a probability distribution through the expected value of a complex exponential. It uniquely determines the distribution and exists for every probability distribution.

Fourier transform: A mathematical transform that represents a function in terms of its frequency components. A characteristic function is the Fourier transform of a probability distribution.
Random variable: A measurable quantity whose value is determined by a random outcome. The distribution encoded by a characteristic function belongs to a random variable.
Central limit theorem: A theorem stating that normalized sums of many independent variables often converge in distribution to a normal law. Characteristic functions provide a standard proof by tracking transforms of normalized sums.
Probability density function: A function whose integral over a set gives the probability that a continuous random variable lies there. A density gives probabilities locally, while a characteristic function encodes the whole distribution in frequency space.
Moment-generating function: A function whose derivatives at zero encode the moments of a probability distribution, when it exists near zero. Unlike this function, it may fail to exist for heavy-tailed distributions.
Probability distribution: A mathematical description of the possible values of a random quantity and their probabilities. The function represents this distribution without requiring a density.
Stable distribution: A probability distribution preserved up to location and scale by sums of independent copies. Its characteristic function has a form that makes this stability under addition explicit.
Probability-generating function: A power series whose coefficients are the probabilities of a nonnegative integer-valued random variable. It is tailored to discrete counts, whereas characteristic functions apply to any real-valued distribution.
Cumulant-generating function: The logarithm of a moment-generating function, whose derivatives at zero give cumulants when defined. Taking a logarithm of a related transform organizes distributional information into cumulants.
Expected value: The probability-weighted average of a random variable's values. The defining formula takes an expected value of a complex exponential.

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