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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 $M_n = \max_{0 \leq i \leq n} S_i$.

  1. What is the distribution of $M_n$ (as a function of $n$)?
  2. 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.

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Do you need small $n$ or only the large $n$ limit? The last one can, probably, be derived from the Brownian motion approximation where you have neat formulae for everything; the exact nature of the steps shouldn't really matter much. –  fedja Feb 20 '13 at 21:30
Ideally, we'd have small $n$ too. But I'd be happy with the large-$n$ limit. –  Jason Cantarella Feb 20 '13 at 23:59
There is a way to solve this problem even for more general Markov processes, but could you please tell me more precise about your conditioning. In your case $\mathsf P(Y_k\geq 0\;\forall k) = 0$ so I guess you are interested rather in $$ \mathsf P(M_n \geq m|Y_k\geq 0 \; \forall 0\leq k\leq n) $$ am I right? –  Ilya Feb 21 '13 at 9:45
@Ilya, you are completely correct (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
I second the fedja's suggestion: to consider the Brownian approximation. Here, your problem is easily solvable: well, you can say that you are comparing the probability of a Brownian path starting from $\eps>0$ to stay positive during the time $n$, and to stay between $0$ and $a$ for the same time. (And pass to the limit as $\esp\to 0$.) This, you can study by the reflection method (what is the same, by studying the behavior of the heat equation with zero boundary condition, what corresponds to an odd extension of the initial condition)... –  Victor Kleptsyn Feb 24 '13 at 21:14

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