(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.
<br />&nbsp;&nbsp;&nbsp;![RandWalk10K][1]<br />
And here is the very same random walk extended to $n=10^5$ steps:
<br />&nbsp;&nbsp;&nbsp;![RandWalk100K][2]


  [1]: https://i.sstatic.net/j4LtC.jpg
  [2]: https://i.sstatic.net/6yYAp.jpg