When does a discrete-time random walk in Rn absorb the origin into its convex hull?

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 Sn.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 BMn(i),i.N, the convex hull of the first Cn 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 ./2-covering time of certain random walks on Sn.1 is of order n. For certain spherical simplices on Sn.1, we prove an extension of Gordon.s theorem dealing with a broad class of random matrices; as an application, we show that Cn steps are sufficient for the standard walk on Zn 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 conv{BMn(t):t.[1,cn]} does not contain the origin with probability close to one.