Given a 2D random walk (in a lattice grid) starting from the origin what is the average waiting time to hit the line $y=1-x$
From my knowledge every point in a 2D random walk has probability one of being traversed in the walk, but trying this problem numerically seems like the expectation is unbounded.
Best Answer
Let $(X_n)$ be the random walk, and $(Y_n)$ the (signed) vertical displacement between $(X_n)$ and the line $y=1-x$, i.e. if $X_n=(x,y)$ then $Y_n=1-x-y$.
Observe that $Y_0=1$, and $$Y_{n+1} = \begin{cases} Y_n+1 & \text{with probability } 1/2, \\ Y_n-1 & \text{with probability } 1/2, \end{cases}$$ So $(Y_n)$ is a simple random walk on $\mathbb Z$. Can you find the expected time for $(Y_n)$ to hit $0$?