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: Consider a queueing system in which K sources each generate 1=M units of work once every time unit. The phases of the sources are mutually independent, and each phase is uniformly distributed over the unit interval. The queue is served at unit rate. The distribution of the typical work and the maximum work are studied both for finite K and M and in the limit of large K and M . The limiting maximum work is identified as the maximum of a reflecting Brownian motion over the unit interval given that its local time at zero first reaches a specified value at the end of the interval. The analysis is expedited by a tie to the empirical process arising in Kolmogorov-Smirnov statistical tests. 1

: This paper presents nonparametric test of independence that can be use to test the independence of p random vectors, serial independence for time series or residuals data. These tests are shown to generalize the classical portmanteau statistics. Applications to both time series and regression residuals are discussed. AMS 1990 subject classifications: Primary 62G10, 60F05, Secondary 62E20. Key words and phrases: independence, serial independence, empirical processes, pseudo-observations, residuals, weak convergence, Cram'er-von Mises statistics. 2 Proposed running head: Nonparametric test of independence Galley proofs should be sent to: Kilani Ghoudi D'epartement de math'ematiques et d'informatique Universit'e du Qu'ebec `a Trois-Rivi`eres Case postale 500 Trois-Rivi`eres (Qu'ebec) Canada G9A 5H7 tel: 1 (819) 376 5170 ex. 3814 fax: 1 (819) 376 5185 e-mail: ghoudi@uqtr.uquebec.ca The final manuscript will be submitted electronically in LaTeX format. 3 1. Introduction Testing f...

Let τ be the number of elements of a sample taken from a population uniformly distributed in [0, 1]. Let α ≥ 0beanumbersuchthatλ = nα ≤ 1. Subdivide an interval of length 1 − λ into n parts by n − 1 independently and uniformly distributed points. Denote δ1,...,δn the lengths of these subintervals. Using the notations Fn(t, λ) =δ1 +...+ δτ + τα, the generalizations of the theorems of Smirnov are expressed by (4.14), (4.15), where Gm(t, μ) is defined similarly to Fn(t, λ) and these two stochastic processes are supposed to be independent. These theorems were already published in [7], the proofs are given here. Applications to inventory problems are also mentioned. 1

We consider a branching random walk for which the maximum position of a particle in the n’th generation, Rn, has zero speed on the linear scale: Rn/n → 0 as n → ∞. We further remove (“kill”) any particle whose displacement is negative, together with its entire descendence. The size Z of the set of un-killed particles is almost surely finite [26, 31]. In this paper, we confirm a conjecture of Aldous [3, 4] that E [Z] < ∞ while E [Z log Z] = ∞. The proofs rely on precise large deviations estimates and ballot theorem-style results for the sample paths of random walks. 1

Gamma n(U n \Gamma 1=2) = n(G n (t) \Gamma t)dt = where G n (t) = n i=1 1 [0;t] (U i ) is the empirical distribution function of the U i 's, and U n (t) = n(G n (t) \Gamma t) is the uniform empirical process (under H). The statistic A was apparently proposed by L. Moses; see Chapman (1958) and Birnbaum and Tang (1964). Under the null hypothesis H the central limit theorem (or CLT) implies that ! N(0; 1=12); i.e. lim n!1 n t) = P (N(0; 1 12 ) t) = \Phi(t 12) for all t 2 R where \Phi(z) = R z \Gamma1 (2) exp(\Gammax =2)dx is the standard normal distribution function. For a finite sample size n, the distribution function of A n can even be calculated exactly; see e.g. Feller (1971), theorem 1a, page 28. Another formulation of the limiting distribution is that U(t)dt N(0; 1=12) ; here U is a standard Brownian bridge process on [0; 1]. This follows from the "functional central limit theorem" for the empirical process U n and the now classic

ForL-estimators, a representation of the jackknife statistic, based on an inherent reverse martingale structure of jackknifing, is incorporated in the study of the asymptotic properties of the estimator as well as the allied jackknife estimator of the standard error. Some applications to sequential analysis are also discussed. 1. Introduction. Let {x 1">l

goodness-of-fit tests using as test statistics the Levy and Prohorov distance between empirical distribution functions. Computational procedures are described for computing the test statistics. Recurrence equations are described for computing the distribution of the two-sample test statistics, using results about the maximal matchings in certain graphs. The asymptotic distribution of the one-sample test statistic is expressed in terms of the distribution of fluctuations in the sample path of the Brownian Bridge stochastic process. Tables of these distributions are given in the appendix. The power of the tests against certain alternatives is discussed, and the results of simulations comparing the power with that of the Kolmogorov

351 cours de la Libération

We consider branching random walks built on Galton–Watson trees with offspring distribution having a bounded support, conditioned to have n nodes, and their rescaled convergences to the Brownian snake. We exhibit a notion of “globally centered discrete snake” that extends the usual settings in which the displacements are supposed centered. We show that under some additional moment conditions, when n goes to +∞, “globally centered discrete snakes ” converge to the Brownian snake. The proof relies on a precise study of the lineage of the nodes in a Galton–Watson tree conditioned by the size, and their links with a multinomial process [the lineage of a node u is the vector indexed by (k,j) giving the number of ancestors of u having k children and for which u is a descendant of the jth one]. Some consequences concerning Galton–Watson trees conditioned by the size are also derived. 1. Introduction.