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(Not an answer; rather an example.)

Here is a random walk with random $\pm 1$ $xy$-steps with equal probability (you didn't specify details), for $n=10^4$ steps, and its convex hull.
   
And here is the very same random walk extended to $n=10^5$ steps:
   

Zakhar Kabluchko, Vladislav Vysotsky, and Dmitry Zaporozhets. "Convex hulls of random walks, hyperplane arrangements, and Weyl chambers." (arXiv abstract.)

"We give an explicit formula for the probability that the convex hull of an $n$-step random walk in $\mathbb{R}^d$ with centrally symmetric density of increments does not contain the origin."

By "contain" here they mean "strictly contain in the interior of the hull."