Random walk processes and their applications in order statistics
This beautifully written paper considers two random walk processes based upon random sampling without replacement, from a collection of \(n\) white and \(n\) black balls. One is the tied-down random walk \(\{\eta_ i\}_{i=0}^{2n}\), where \(\eta_ i\) denotes the difference between the number of white balls and the number of black balls among the first \(i\) balls drawn \((\eta_ 0=\eta_{2n}=0)\). If we let \(\Omega^ +\) denote the set of those sample points \(\omega\) such that \(\eta_ i(\omega)\geq 0\) for all \(i\), if we treat \(\Omega^ +\) as a new sample space with each of its \((n+1)^{-1}{2n \choose n}\) points equally likely, and if we let \(\eta_ i^ +\) denote the restriction of \(\eta_ i\) to \(\Omega^ +\), then we obtain the Bernoulli excursion \(\{\eta_ i^ +\}_{i=0}^{2n}\). The study focuses on three statistics: \(\omega_ n=\text{ the mean of the }\eta_ i^ +\), \(\rho_ n=\) the difference between the mean and the minimum of the \(\eta_ i\), and \(\sigma_ n=\) the mean of the \(|\eta_ i|\). The distributions and moments of these statistics are obtained via a generating function approach, and their asymptotic behaviour is determined. In particular, if \(\{\eta(t):\;0\leq t\leq 1\}\) denotes a special Gaussian process called the tied down Brownian motion process, then the asymptotic distributions of both \(\omega_ n/\sqrt {2n}\) and \(\rho_ n/\sqrt {2n}\) coincide with the distribution of the random variable \[ \rho=\int_ 0^ 1 \eta(t)dt-\min_{0\leq t\leq 1}\eta(t), \] while that of \(\sigma_ n/\sqrt {2n}\) coincides with the distribution of \(\sigma=\int_ 0^ 1|\eta(t)| dt\). Both of these distributions are described in considerable detail. Finally, two statistics (\(\Theta_ n\) and \(\Delta_ n\)) to test whether the elements of two independent samples of size \(n\) have a common distribution are shown to have the same asymptotic behaviour as \(\rho_ n/\sqrt {2n}\) and \(\sigma_ n/\sqrt {2n}\), respectively. We might ask whether the ``min'' expressions in formulas (100) and (128) are not always just \(=0\).
deviation between empirical distributions
scientific article; zbMATH DE number 90047