LévyKhinchine

The Lévy-Khinchine formula is a fundamental result in probability theory that characterizes all infinitely divisible probability distributions on the real line. An infinitely divisible distribution is one that can be represented as the distribution of the sum of an arbitrary number of independent and identically distributed random variables. The formula states that the characteristic function of any infinitely divisible distribution can be expressed in a specific form involving a drift term, a variance term, and a Lévy measure.

The characteristic function, denoted by $\phi(t)$, is given by $\phi(t) = E[e^{itX}]$, where $X$ is a random variable and $i$ is the imaginary unit. The Lévy-Khinchine formula for a probability distribution $\mu$ on $\mathbb{R}$ states that its characteristic function is of the form:

$\log \phi(t) = i\gamma t - \frac{1}{2}\sigma^2 t^2 - \int_{-\infty}^{\infty} (1 - e^{itu} + itu) \nu(du)$

where $\gamma$ is a real number representing the drift, $\sigma^2 \ge 0$ is a real number representing the variance of the Gaussian component, and $\nu$ is a non-negative measure on $\mathbb{R} \setminus \{0\}$ called the Lévy measure. The Lévy measure describes the jump behavior of the corresponding Lévy process. The integral part of the formula accounts for the contribution of jumps. The condition $\int_{-\infty}^{\infty} \min(u^2, 1) \nu(du) < \infty$ is necessary for the integral to converge. This formula is crucial for the study of stochastic processes with independent and stationary increments, known as Lévy processes.