LévyKhintchineesityksen

The Lévy-Khinchine theorem is a fundamental result in probability theory that characterizes the set of 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 theorem states that a probability distribution on the real line is infinitely divisible if and only if its characteristic function can be expressed in a specific form, known as the Lévy-Khinchine representation. This representation involves three components: a drift term, a Lévy measure, and a variance term. The characteristic function of an infinitely divisible distribution $X$ is given by:

$\phi_X(t) = \exp\left(it\gamma - \frac{1}{2}\sigma^2 t^2 + \int_{-\infty}^{\infty} (e^{ixt} - 1 - ixt) \frac{1}{|x|^2} \nu(dx)\right)$

where $\gamma$ is a real number representing the drift, $\sigma^2$ is a non-negative real number representing the variance, and $\nu$ is a non-negative measure on $\mathbb{R} \setminus \{0\}$ satisfying certain integrability conditions, known as the Lévy measure. The integral term captures the contribution of compound Poisson processes and other jump-like phenomena.

The Lévy-Khinchine theorem is crucial for understanding a wide class of probability distributions beyond the well-known Gaussian and Poisson distributions. It forms the theoretical basis for the study of Lévy processes, which are stochastic processes with stationary and independent increments, and have applications in areas such as finance, physics, andQueueing theory.