Universal Laws for the Times of the Maximum and Minimum of Random Walks

Let $t_{\max}$ and $t_{\min}$ be the times of the maximum and the minimum of a random walk of $n$ steps, and $\tau = t_{\min} – t_{\max}$. Mori, Majumdar and Schehr conjectured that $P(\tau = n) = 1/(2n)$ for every symmetric continuous jump law and proved it for $n \le 3$ and for double-exponential jumps. We prove it for every $n$ and every i.i.d. symmetric law without atoms. The proof is the cycle lemma applied to the doubled sequence $(\xi_1, \dots, \xi_n, -\xi_1, \dots, -\xi_n)$, whose cyclic rotations are the doubled sequences of $T^r\xi$, $T(x_1, \dots, x_n) = (x_2, \dots, x_n, -x_1)$: for any law invariant under $T$ without ties, dependent laws included, the highest point of the doubled loop is uniform on its $2n$ positions. These positions are explicit events about $t_{\max}$, $t_{\min}$ and the sign of $M + m – S_n$, each of probability $1/(2n)$. For i.i.d. steps the rest of the joint law depends on the step law through one probability, which gives the universal bound $P(\tau = j) \le 1/(2j)$, the limit $n P(\tau = n – k) \to 1/2$ and, for every stable domain of attraction, the Lévy-flight edge value $1/2$ in averaged form. Brownian motion has a uniform maximizer of the doubled path on $[0, 2)$; on the lattice a uniform split of ties restores the law. Exact enumeration for $n \le 7$ and Monte Carlo over ten step laws, within $4.34$ standard errors, confirm the results.