Maximum final component of a simple symmetric multidimensional random walk?

[Saketh]

I am developing a simple probabilistic model of my own mistakes in (1) solving math problems and (2) implementing algorithms on a computer. I have reduced the problem to one which seems simple enough, but which I have been unable to solve, due to my mathematical inexperience. I figured I would ask for help!

Q: What is the probability that the maximum component at the end of a simple symmetric random walk of length $n$ on $\mathbb{Z}^d$ is greater than $k$?

More formally, I am looking for
$$\mathbb{P}(\operatorname{max}\{S_r \cdot \hat{\textbf{e}}_i \,:\, r \in \{1, \dots, n\}, \,i \in \{1, \dots, d\}\} > k)$$ where the formalization of the random walk $S_n$ and the standard basis $\hat{\textbf{e}}_1, \dots, \hat{\textbf{e}}_m$ is, to the best of my ability, following the notation from these two resources which I found:

• http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/Chapter12.pdf
• http://www.math.uchicago.edu/~lawler/srwbook.pdf

 

[Valkarie]

In other words, I want to know the probability that the maximum component of a symmetric random walk on \mathbb{Z}^d is greater than k at step n. Any help would be greatly appreciated.