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

V. V. Vysotsky

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 48-67

Résumé

Let $S_i$ be a random walk with standard exponential increments. We denote by $\sum_{i=1}^k S_i$ its $k$-step area. The random variable $\inf_{k\ge 1}\frac2{k(k+1)}\sum_{i=1}^k S_i$ plays important role in the study of so-called one-dimensional sticky particles model. We find the distribution of this variable and prove that for $0\le t\le 1$, $$ \mathbf P\,\biggl\{\inf_{k\ge 1}\frac2{k(k+1)}\sum_{i=1}^k S_i\ge t\biggr\}=\mathbf P\,\biggl\{\inf_{k\ge 1}\sum_{i=1}^k\bigl(S_i-it\bigr)\ge 0\biggr\}=\sqrt{1-t}\,e^{-t/2} $$ We also show that for $0\le t\le 1$, $$ \lim_{n\to\infty}\,\mathbf P\,\biggl\{\min_{1\le k\le n}\frac{2n}{k(k+1)}\sum_{i=1}^k U_{i,n}\ge t\biggr\}=\sqrt{1-t}\,e^{-t/2}, $$ where $U_{i, n}$ are the order statistics of $n$ i.i.d. random variables uniformly distributed on $[0,1]$.