Ladder variables and Sparre-Andersen transforms

Consider a centered random walk $S_k$. What is the probability $p_n$ that its first $n$ steps are positive? It is amazing that if the walk is symmetric and distributed continuously, then $p_n$ do not depend on the distribution of increments! In the end of 1940s Sparre-Andersen proved this results with analytic methods but later in the 1960s Feller found an elegant combinatorial explanation based on cyclic transforms of the trajectory of the walk. Similar transforms are used to prove quite a number of statements, for example, the arcsine law for general symmetric continuously distributed walks or to find the probability that the trajectory of the walk lies below the chord joining the origin and $(n, S_n).$ The later result is extremely useful for finding the distribution of the number of segments in the convex minorant of a walk. We will also discuss the idea of Alili and Doney (1999) who generalized Feller's method and found a simple and very useful formula for the joint distribution of the first ladder variables of the walk, that is the moment of the first jump to negative half-axis and the size of this jump. We will show some limit theorems that illustrate the use of this formula.