Explain why the finite-dimensional distributions of a Levy process are completely determined by the distribution of ´Z(t), that is, the one-dimensional distributions. Let Z be a Levy processes. Let  ...

Explain why the finite-dimensional distributions
of a Levy process are completely determined by the distribution of ´Z(t), that is, the one-dimensional distributions.

Let Z be a Levy processes. Let

be the Laplace transform of Z(1). Prove that for any rational number r ≥ 0, the Laplace transform of Z(r) is
  Use this to prove the general expression for the Laplace transform of Z(t) given in (11.6).