When does a discrete-time random walk in $\mathbb{R}^{n}$ absorb the origin into its convex hull?

Abstract

We connect this question to a problem of estimating the probability that the image of certain random matrices does not intersect with a subset of the unit sphere $\mathbb{S}^{n-1}$. In this way, the case of a discretized Brownian motion is related to Gordon’s escape theorem dealing with standard Gaussian matrices. We show that for the random walk $\mathrm{BM}_{n}(i),i\in\mathbb{N}$, the convex hull of the first $C^{n}$ steps (for a sufficiently large universal constant $C$) contains the origin with probability close to one. Moreover, the approach allows us to prove that with high probability the $\pi/2$-covering time of certain random walks on $\mathbb{S}^{n-1}$ is of order $n$. For certain spherical simplices on $\mathbb{S}^{n-1}$, we prove an extension of Gordon’s theorem dealing with a broad class of random matrices; as an application, we show that $C^{n}$ steps are sufficient for the standard walk on $\mathbb{Z}^{n}$ to absorb the origin into its convex hull with a high probability. Finally, we prove that the aforementioned bound is sharp in the following sense: for some universal constant $c>1$, the convex hull of the $n$-dimensional Brownian motion $\operatorname{conv}\{\mathrm{BM}_{n}(t):t\in[1,c^{n}]\}$ does not contain the origin with probability close to one.