# hitting probabilities of oriented random walk

Consider a random walk on $$\mathbb{Z}^2$$, starting at $$(0,0)$$. Each step it moves rightwards with probability $$p$$ and upwards with probability $$q=1-p$$. The random walk terminates when it hits the right boundary $$x=T$$ or top boundary $$y=T$$ for some threshold $$T$$. The question is to find the probability that the random walks hit the right boundary and the probability that it hits the top boundary.

I feel this may be a classical problem but I cannot find any reference. It is easy to see that $$\Pr\{\text{hitting right boundary}\} = \sum_{i=0}^T \binom{i+T}{T} q^i p^{T+1}$$ and $$\Pr\{\text{hitting top boundary}\} = \sum_{i=0}^T \binom{i+T}{T} p^i q^{T+1}.$$ There seems no simple closed expression for the probabilities.

Question 1. Is there a simple proof to show that the two probabilities above sum up to $$1$$?

Question 2. Consider the random walk in a $$d$$-dimensional space, starting at $$x=0$$. Each time we have $$x\gets x+e_i$$ with probability $$p_i$$, where $$e_1,\dots,e_d$$ are the canonical basis vectors and $$\sum_i p_i=1$$. The random walk terminates when $$\|x\|_\infty=T$$. Similarly we can express the probability $$\Pr\{x_i = T\}$$ in terms of multinomial coefficients.

What is an asymptotic upper bound for $$\Pr\{x_i = T\}$$ for each $$i$$ upon termination?

I feel these are probably standard questions so perhaps somebody can point me to literature.