On the Lengths of Curves Passing through Boundary Points of a Planar Convex Shape

We study the lengths of curves passing through a fixed number of points on the boundary of a convex shape in the plane. We show that for any convex shape $K$, there exist four points on the boundary of $K$ such that the length of any curve passing through these points is at least half of the perimeter of $K$. It is also shown that the same statement does not remain valid with the additional constraint that the points are extreme points of $K$. Moreover, the factor $\frac12$ cannot be achieved with any fixed number of extreme points. We conclude the paper with few other inequalities related to the perimeter of a convex shape.Comment: 7 pages, 8 figure

Positivity of integrated random walks

Abstract. Take a centered random walk Sn and consider the sequence of its partial sums An: = ∑n i=1 Si. Suppose S1 is in the domain of normal attraction of an α-stable law with 1 < α ≤ 2. Assuming that S1 is either right-exponential (that is P(S1> x|S1> 0) = e−ax for some a> 0 and all x> 0) or right-continuous (skip free), we prove tha

Convex hulls of multidimensional random walks

Let Sk be a random walk in R d such that its distribution of increments does not assign mass to hyperplanes. We study the probability pn that the convex hull conv(S1, . . . , Sn) of the first n steps of the walk does not include the origin. By providing an explicit formula, we show that for planar symmetrically distributed random walks, pn does not depend on the distribution of increments. This extends the well known result by Sparre Andersen (1949) that a one-dimensional random walk satisfying the above continuity and symmetry assumptions stays positive with a distribution-free probability. We also find the asymptotics of pn as n → ∞ for any planar random walk with zero mean square-integrable increments. We further developed our approach from the planar case to study a wide class of geometric characteristics of convex hulls of random walks in any dimension d ≥ 2. In particular, we give formulas for the expected value of the number of faces, the volume, the surface area, and other intrinsic volumes, including the following multidimensional generalization of the Spitzer–Widom formula (1961) on the perimeter of planar walks: EV1(conv(0, S1, . . . , Sn)) = Xn k=1 ΣkSkk k, where V1 denotes the first intrinsic volume, which is proportional to the mean width. These results have applications to geometry, and in particular, imply the formula by Gao and Vitale (2001) for the intrinsic volumes of special path-simplexes, called canonical orthoschemes, which are finite-dimensional approximations of the closed convex hull of a Wiener spiral. Moreover, there is a direct connection between spherical intrinsic volumes of these simplexes and the probabilities pn. We also prove similar results for convex hulls of random walk bridges, and more generally, for partial sums of exchangeable random vectors

Stability of overshoots of zero mean random walks

We prove that for a random walk on the real line whose increments have zero mean and are either integer-valued or spread out (i.e. the distributions of steps of the walk are eventually non-singular), the Markov chain of overshoots above a fixed level converges in total variation to its stationary distribution. We find the explicit form of this distribution heuristically and then prove its invariance using a time-reversal argument. If, in addition, the increments of the walk are in the domain of attraction of a non-one-sided α-stable law with index α∈(1,2) (resp. have finite variance), we establish geometric (resp. uniform) ergodicity for the Markov chain of overshoots. All the convergence results above are also valid for the Markov chain obtained by sampling the walk at the entrance times into an interval

On the completion of Skorokhod space

We consider the classical Skorokhod space D[0,1] and the space of continuous functions C[0,1] equipped with the standard Skorokhod distance ρ. It is well known that neither (D[0,1],ρ) nor (C[0,1],ρ) is complete. We provide an explicit description of the corresponding completions. The elements of these completions can be regarded as usual functions on [0,1] except for a countable number of instants where their values vary “instantly"