Consider $m$ iid symmetric random walks on the integers that start at the origin. Let $D^i_x$ be the distance of the $i$th walk from the origin after $x$ steps. For a given $x$, what is $E[\min_{1\leq i \leq m} D_x^i]$, that is, what is the minimum distance of a random walk from the origin from among the $m$ walks? Is there a nice asymptotic approximation for this expression?
2026-04-06 21:15:10.1775510110
Minimum displacement of iid random walks
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