Representation of linearly additive random fields

Summary Chentsov type representation theorem is proved for stochastically continuous, linearly additive, infinitely divisible random field without Gaussian component, where a random fieldX={X(t), tR ^d } is called linearly additive if the stochastic process defined by ()=X(a+b), R, has independent increments for every pair(a, b), a, bR ^d . In passing it is shown that there exists a natural one-to-one correspondence between stochastically continuous, linearly additive Poisson random fields onR ^d and locally finite, bundleless measures on the space of all (d-1)-hyperplanes inR ^d . The latter result is closely related to Ambartzumian's theorem on the representation of linearly additive pseudometrics in the plane.