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

Authors:

V V Vysotsky

Institution:

(1) St.Petersburg State University, St.Petersburg, Russia

Abstract:

Let S_i be a random walk with standard exponential increments. The sum ∑ _i=1 ^k S_i is called the k-step area of the walk. The random variable $\mathop {\inf }\limits_{k \geqslant 1} \tfrac{2}{{k(k + 1)}}$ ∑ _i=1 ^k S_i plays an important role in the study of the so-called one-dimensional sticky particles model. We find the distribution of this variable and prove that

$\mathbb{P}\left\{ {\mathop {\inf }\limits_{k \geqslant 1} \tfrac{2}{{k(k + 1)}}\sum\limits_{i = 1}^k {S_i } \geqslant t} \right\} = \mathbb{P}\left\{ {\mathop {\inf }\limits_{k \geqslant 1} \sum\limits_{i = 1}^k {(S_i - it)} \geqslant 0} \right\} = \sqrt {1 - t} e^{ - t/2}$

for 0 ≤ t ≤ 1. We also show that

$\mathop {\lim }\limits_{n \to \infty } \mathbb{P}\left\{ {\mathop {\min }\limits_{1 \leqslant k \leqslant n} \tfrac{{2n}}{{k(k + 1)}}\sum\limits_{i = 1}^k {U_{i,n} } \geqslant t} \right\} = \sqrt {1 - t} e^{ - t/2}$

, where the U_i,n are order statistics of n i.i.d. random variables uniformly distributed on 0, 1]. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 48–67.

Keywords:

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