We start by revising basic concepts of points processes (PP) and random measures in a general space: their definition, distribution, probability generating functional,Palm distribution. We describe the main classes of PPs: Binomial, Poisson, Gibbs, cluster processes, branching random walk. Next, we establish specific properties of the Poisson PP and consider a few applications based on the Poisson PP models in statistics and telecommunications. We then pass to the limit theorems involving superposition of PPs and characterise possible limits: infinitely divisible, self-decomposable and stable PPs. If time allows, we finish by simulation methods and statistical inference.

In these lectures, I plan to cover two topics:

• The basic theory of Levy processes (or processes with stationary and independent increments). This class of stochastic processes originated in the theory of addition of random variables (Kolmogorov, Gnedenko, Levy, among others). I will include an exposition of the Wiener-Hopf factorization, the theory of hitting times and fluctuation. I will also discuss some applications, depending on time and interest. I will give emphasis to probabilistic proofs, as much as possible.
  
• The theory of symmetry and exchangeability. I will review some concepts of ergodic theory and proceed to show the Ryll-Nardzewski approach to de Finetti's theorem. The theory is fundamental to the foundation of Bayesian statistics. I will also include a discussion of finite exchangeability and symmetry.