LévyKhintchineformeln

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 n independent and identically distributed random variables for any positive integer n. This means that such a distribution can be "divided" into any number of independent and identical parts.

The theorem states that a probability distribution is infinitely divisible if and only if its characteristic function can be expressed in a specific form. This form is known as the Lévy-Khinchine representation. The representation involves three components: a drift term, a Gaussian component, and a Lévy measure. The drift term accounts for the expected value of the underlying process. The Gaussian component is related to the variance of the process and corresponds to a normal distribution. The Lévy measure, often denoted by $\nu$, is a non-negative measure on the real line that describes the jump behavior of the process. It governs the probability and size of jumps that occur.

The Lévy-Khinchine formula provides a concrete way to construct and identify infinitely divisible distributions. It is a powerful tool for studying stochastic processes, particularly Lévy processes, which are stochastic processes with stationary and independent increments. These processes generalize Brownian motion by allowing for jumps. The formula is central to the study of probability measures and their properties, and it has significant applications in areas such as financial mathematics, physics, and signal processing.