# Where does directed random walk hit the boundary of a region?

I have a problem that I more or less know the answer to (in an ad hoc way), but would really like to see it done in a systematic way. In spite of this, I will pose the question in quite a concrete way. Note: this question was previously posted on stackexchange.

Consider a two-dimensional random walk on $\mathbb Z^2$. Fix a finite subset $S$ of $\mathbb Z^2$ in which each element of $S$ has strictly positive $x$-coordinate and assign a probability measure $\mu$ to $S$ Write the location of the 2D walk as $(X_n,Y_n)$ and let $(X_{n+1},Y_{n+1})=(X_n,Y_n)+(u,v)$, where $(u,v)$ is a randomly chosen element of $S$, chosen with distribution $\mu$.

Let $T=\inf\lbrace n\colon X_n\ge M\rbrace$ for some (large) fixed $M$. I'm looking for a way to describe $Y_T$.

Here's what I think is the answer: Write $(U,V)$ for a random element of $S$, write $\bar U=\mathbb EU$ and $\bar V=\mathbb EV$ (here $\mathbb E$ is with respect to $\mu$). I expect that $Y_T$ will have a distribution (for large $M$) close to a normal distribution with mean $M\bar V/\bar U$ and variance $(M/\bar U)\mathbb E(V-U\bar V/\bar U)^2$.

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