Geodesic
The area of exponential random walk and partial sums of uniform order statistics
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 48-67
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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 $[0,1]$. 
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     author = {V. V. Vysotsky},
     title = {The area of exponential random walk and partial sums of uniform order statistics},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {48--67},
     year = {2007},
     volume = {341},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a2/}
}
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V. V. Vysotsky. The area of exponential random walk and partial sums of uniform order statistics. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 11, Tome 341 (2007), pp. 48-67. http://geodesic.mathdoc.fr/item/ZNSL_2007_341_a2/

[1] V. V. Vysotskii, “Energiya i kolichestvo klasterov v stokhasticheskikh sistemakh neuprugikh prityagivayuschikhsya chastits”, Teoriya veroyatn. i ee primen., 50 (2005), 241–265 | MR

[2] S. Karlin, Osnovy teorii sluchainykh protsessov, Mir, M., 1971 | MR | Zbl

[3] C. Giraud, “Clustering in a self-gravitating one-dimensional gas at zero temperature”, J. Stat. Phys., 105 (2001), 585–604 | DOI | MR | Zbl

[4] M. Lifshits and Z. Shi, “Aggregation rates in one-dimensional stochastic systems with adhesion and gravitation”, Ann. Probab., 33 (2005), 53–81 | DOI | MR | Zbl

[5] Ph. A. Martin and J. Piasecki, “Aggregation dynamics in a self-gravitating one-dimensional gas”, J. Stat. Phys., 84 (1996), 837–857 | DOI | MR | Zbl