Abstract:
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.
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