The area of exponential random walk and partial sums of uniform order statistics

Details The area of exponential random walk and partial sums of uniform order statistics The area of exponential random walk and partial sums of uniform order statistics

2006-12-17

arxiv.org/abs/math/0612490v2

Mathematics

Probability

Added one reference; corrected few typos; added one sentence to Proposition 1

Scientific paper

Let S_i be a random walk with standard exponential increments. We call \sum_{i=1}^k S_i its k-step area. The random variable V = \inf_{k \ge 1} \frac{2}{k(k+1)} \sum_{i=1}^k S_i plays important role in the study of so-called one-dimensional sticky particles model. We find the distribution of V and prove that P(V > t) = \sqrt{1-t} exp(-t/2) for t in [0,1]. We also show that the variables \min_{1 \le k \le n} \frac{2n}{k(k+1)} \sum_{i=1}^k U_{i, n} converge in distribution to V. Here U_{i, n} are the order statistics of n i.i.d. random variables uniformly distributed on [0,1].

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