Let $X^1$ and $X^{-1}$ be two simple random walk in $\mathbb{Z}$ starting respectively from $1$ and $-1$. Let $\tau$ be the first time one of them reaches the origin,
$$\tau = \inf \{ j \geq 0 \, : \, X^{-1}(j) = 0 \, \, \mbox{or}\, \, X^{1}(j) = 0 \}.$$
How much is $E[\tau]$, the expectation of $\tau$?
Comment: If there was only a random walk, then the expectation of $\tau$ would be infinite, as $P(\tau >k ) \sim 1/k$. However, for two random walks I think it should not be infinite, as $P(\tau >k ) \sim 1/k^2$. How to make it rigorous and compute the exact number?