distribution of a Lévy process has the property of infinite divisibility: given any integer n, the law of a Lévy process at time t can be represented as the law of n independent random variables, which are precisely the increments of the Lévy process over time intervals of length t/n, which are independent and identically distributed by assumptions 2 and 3. Conversely, for each infinitely divisible probability distribution , there is a Lévy process such that the law of is given by . Moments In any Lévy process with finite moments, the nth moment , is a polynomial function of t; these functions satisfy a binomial identity: Lévy–Khintchine representation The distribution of a Lévy process is characterized by its characteristic function, which is given by the Lévy–Khintchine formula (general for all infinitely divisible distributions): If is a Lévy process, then its characteristic function is given by where , , and is a -finite measure called the Lévy measure of , satisfying the property In the above, is the indicator function. Because characteristic functions uniquely determine their underlying probability distributions, each Lévy process is uniquely determined by the "Lévy–Khintchine triplet" . The terms of this triplet suggest that a Lévy process can be seen as having three independent components: a linear drift, a Brownian motion, and a Lévy jump process, as described below. This immediately gives that the only (nondeterministic) continuous Lévy process is a Brownian motion with drift; similarly, every Lévy process is a semimartingale. Lévy–Itô decomposition Because the characteristic functions of independent random variables multiply, the Lévy–Khintchine theorem suggests that every Lévy process is the sum of Brownian motion with drift and another independent random variable, a Lévy jump process. The Lévy–Itô decomposition describes the latter as a (stochastic) sum of independent Poisson random variables. Let — that is, the restriction of to , renormalized to be a probability measure; similarly, let (but do not rescale). Then The former is the characteristic function of a compound Poisson process with intensity and child distribution . The latter is that of a compensated generalized Poisson process (CGPP): a process with countably many jump discontinuities on every interval a.s., but such that those discontinuities are of magnitude less than . If , then the CGPP is a pure jump process. Therefore in terms of processes one may decompose in the following way where is the compound Poisson process with jumps larger than