Universal first-passage properties of discrete-time random walks and Lévy flights on a line: Statistics of the global maximum and records

In these lecture notes I will discuss the universal first-passage properties of a simple correlated discrete-time sequence {x0=0,x1,x2,…,xn}{x0=0,x1,x2,…,xn} up to nn steps where xixi represents the position at step ii of a random walker hopping on a continuous line by drawing independently, at each time step, a random jump length from an arbitrary symmetric and continuous distribution (it includes, e.g., the Lévy flights). I will focus on the statistics of two extreme observables associated with the sequence: (i) its global maximum and the time step at which the maximum occurs and (ii) the number of records in the sequence and their ages. I will demonstrate how the universal statistics of these observables emerge as a consequence of Pollaczek–Spitzer formula and the associated Sparre Andersen theorem.