Let $\tau_1$, $\tau_2$ be two nonnegative integer valued random variables such that $P( \tau_1 > r) \leq P( \tau_2 > r)$ for any $r \geq 0$, let $X(i)$ be a symmetric random walk on $\mathbb{Z}$ starting from the origin which is independent from $\tau_1, \tau_2$.
How to prove that $$ P( | X( \tau_1 )| \geq r ) \leq P( | X( \tau_2 )| \geq r ) $$ for any $r \geq 0$? Namely, if we wait longer time, the random walk is less localized around its starting point.