arXiv subjects

Vladislav Vysotsky

Publications and source records attributed to Vladislav Vysotsky.

25 records · Page 2Linked to original sources

arXiv2016

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.

math.MG↗

arXiv2015

On the weak limit law of the maximal uniform k-spacing

This paper gives a simple proof of a limit theorem for the lenght of the largest interval straddling a fixed number of i.i.d. points uniformly disributed on a unit interval. The key step in our argument is a classical theorem of Watson (1954) on the maxima of m-dependent stationary stochastic sequences.

math.PR↗

arXiv2014

Limit theorems for random walks that avoid bounded sets, with applications to the largest gap problem

Consider a centred random walk in dimension one with a positive finite variance $σ^2$, and let $τ_B$ be the hitting time for a bounded Borel set $B$ with a non-empty interior. We prove the asymptotic $P_x(τ_B > n) \sim \sqrt{2 / π} σ^{-1} V_B(x) n^{-1/2}$ and provide an explicit formula for the limit $V_B$ as a function of the initial position $x$ of the walk. We also give a functional limit theorem for the walk conditioned to avoid $B$ by the time $n$. As a main application, consider the case that $B$ is an interval and study the size of the largest gap $G_n$ (maximal spacing) within the range of the walk by the time $n$. We prove a limit theorem for $G_n$, which is shown to be of the constant order, and describe its limit distribution. In addition, we prove an analogous result for the number of non-visited sites within the range of an integer-valued random walk.

math.PR↗

arXiv2012

Positivity of integrated random walks

Take a centered random walk S_n and consider the sequence of its partial sums A_n = S_1 + ... + S_n. Suppose S_1 is in the domain of normal attraction of an α-stable law with 1 < α<= 2. Assuming that S_1 is either right-exponential (that is P(S > x | S > 0)=e^{-ax} for some a > 0 and all x > 0) or right-continuous (skip free), we prove that p_N = P(A_1 > 0, ..., A_N > 0) ~ C_αN^{1/(2α) - 1/2} as N tends to infinity, where C_α> 0 depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

math.PR↗

arXiv2010

On the probability that integrated random walks stay positive

Let $S_n$ be a centered random walk with a finite variance, and define the new sequence $A_n:=\sum_{i=1}^n S_i$, which we call an integrated random walk. We are interested in the asymptotics of $$p_N:=P(\min_{1 \le k \le N} A_k \ge 0)$$ as $N \to \infty$. Sinai (1992) proved that $p_N \asymp N^{-1/4}$ if $S_n$ is a simple random walk. We show that $p_N \asymp N^{-1/4}$ for some other types of random walks that include double-sided exponential and double-sided geometric walks, both not necessarily symmetric. We also prove that $p_N \le c N^{-1/4}$ for lattice walks and for upper exponential walks, that are the walks such that $Law (S_1 | S_1>0)$ is an exponential distribution.

math.PR↗

arXiv2007

The area of exponential random walk and partial sums of uniform order statistics

Let S_i be a random walk with standard exponential increments. We call \sum_{i=1}^k S_i its k-step area. The random variable V = \inf_{k \ge 1} \frac{2}{k(k+1)} \sum_{i=1}^k S_i plays important role in the study of so-called one-dimensional sticky particles model. We find the distribution of V and prove that P(V > t) = \sqrt{1-t} exp(-t/2) for t in [0,1]. We also show that the variables \min_{1 \le k \le n} \frac{2n}{k(k+1)} \sum_{i=1}^k U_{i, n} converge in distribution to V. Here U_{i, n} are the order statistics of n i.i.d. random variables uniformly distributed on [0,1].

math.PR↗

arXiv2006

A functional limit theorem for the position of a particle in a Lorentz type model

Consider a particle moving through a random medium, which consists of spherical obstacles, randomly distributed in R^d. The particle is accelerated by a constant external field; when colliding with an obstacle, the particle inelastically reflects. We study the asymptotics of X(t), which denotes the position of the particle at time t, as t tends to infinity. The result is a functional limit theorem for X(t).

math.PR↗