Abstract:
It is well known that the probability that a random walk stays
positive by the time $n$ does not depend on the distribution of the
random walk if it is continuous and symmetric. We generalize this
result to dimension two and suggest a conjecture for arbitrary
dimension. We also derive a formula for the expected perimeter and
number of faces of the convex hull of a multidimensional random walk
with arbitrary continuous distribution. It turned out that the
expected number of faces does not depend on the distribution, too.
Based on the joint work with V. Vysotsky.