ABSTRACT
The fundamental theorem of the theory of processes with (stationary) independent increments states that if such a process is stochastically continuous. By the assumed independence, the joint distribution of the increments is the product of their marginal distributions. A major theorem of probability theory furnishes the canonical form of the characteristic function of an infinitely divisible distribution. The marginal distributions of the increments uniquely determine the joint distributions of the process under the condition X0 = 0. A classical result is that the distribution of the increment Xt - Xs of a process with independent increments is infinitely divisible. When the increments are symmetrically distributed about 0 in addition to being stationary and independent, the imaginary part of the function in the representation is dropped.
