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Maximum final component of a simple symmetric multidimensional random walk?

  1. Apr 3, 2012 #1
    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 [itex]n[/itex] on [itex]\mathbb{Z}^d[/itex] is greater than [itex]k[/itex]?

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