I have an $n$ step random walk which starts at zero $X_0 = 0 = S_0$ where the steps $X_i$ are independent uniform random variates in $[-1,1]$, but the walk is conditioned on the hypothesis that it stays in the right half-line $[0,\infty)$ (that is, $S_k = X_0 + X_1 + \dots + X_k \geq 0$ for $k \in 0, \dots, n$).

I'm interested in the maximum of the walk, or the random variable $M_n = \max_{0 \leq i \leq n} S_i$.

- What is the distribution of $M_n$ (as a function of $n$)?
- Can one derive (upper or lower) bounds on the probability $P[M_n < a]$ that the walk is actually contained in the interval $[0,a]$ instead of $[0,\infty)$? Both types of bounds would be interesting, for different reasons, in a polymer physics question.

I've looked at the classical references (eg. Feller, or Spitzer, Chapter 4), but Spitzer at least seems to deal only with the integer valued case. However, it seems that I'm either missing it, or I can't quite find the right reference.

Are these standard results with a standard reference? It's clearly related to A random walk with uniformly distributed steps, which deals with the overall volume of the space of walks with uniform random variates as steps conditioned to stay positive, but not the maximum.

forehead smack). I edited the question to make it more clear that I'm thinking about the max of a finite walk which is conditioned to stay in $[0,\infty)$ for $n$ steps. – Jason Cantarella Feb 21 '13 at 21:21