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26
A Queue with Periodic Arrivals and Constant Service Rate
 In Chapter 10 of Probability, Statistics and Optimization a Tribute
, 1994
"... : 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 max ..."
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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 KolmogorovSmirnov statistical tests. 1
A Nonparametric Test Of Serial Independence For Time Series And Residuals
"... : 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 resi ..."
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: 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, pseudoobservations, residuals, weak convergence, Cram'ervon 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 TroisRivi`eres Case postale 500 TroisRivi`eres (Qu'ebec) Canada G9A 5H7 tel: 1 (819) 376 5170 ex. 3814 fax: 1 (819) 376 5185 email: ghoudi@uqtr.uquebec.ca The final manuscript will be submitted electronically in LaTeX format. 3 1. Introduction Testing f...
Empirical processes in probabilistic number theory: the LIL for the discrepancy of (nkω) mod 1
"... ..."
Total Progeny in Killed Branching Random Walk
, 2009
"... 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 u ..."
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Cited by 3 (1 self)
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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 unkilled 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 theoremstyle results for the sample paths of random walks. 1
Generalizations of the theorems of Smirnov with application to a reliability type inventory problem, Mathematische Operationsforschung und Statistik 4
, 1973
"... 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. U ..."
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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
An exposition of Bretagnolle and Massart’s proof of the KMT theorem for the uniform empirical process
, 2005
"... empirical process defined in terms of empirical distribution functions. A proof, expanding on one in a 1989 paper by Bretagnolle and Massart, is given for the Koml ´ osMajorTusn ´ ady result on the speed of convergence of the empirical process to a Brownian bridge in the supremum norm. ..."
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empirical process defined in terms of empirical distribution functions. A proof, expanding on one in a 1989 paper by Bretagnolle and Massart, is given for the Koml ´ osMajorTusn ´ ady result on the speed of convergence of the empirical process to a Brownian bridge in the supremum norm.
New Techniques for Empirical Processes of Dependent Data
, 2008
"... We present a new technique for proving empirical process invariance principle for stationary processes (Xn)n≥0. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions (f(Xn))n≥0, not containing the indic ..."
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We present a new technique for proving empirical process invariance principle for stationary processes (Xn)n≥0. The main novelty of our approach lies in the fact that we only require the central limit theorem and a moment bound for a restricted class of functions (f(Xn))n≥0, not containing the indicator functions. Our approach can be applied to Markov chains and dynamical systems, using spectral properties of the transfer operator. Our proof consists of a novel application of chaining techniques. 1
THE LINEAGE PROCESS IN GALTON–WATSON TREES AND GLOBALLY CENTERED DISCRETE SNAKES
, 2008
"... 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 ..."
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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.
Maximal inequality for highdimensional cubes
 Confluentes Mathematici
"... Abstract. We present lower estimates for the best constant appearing in the weak (1, 1) maximal inequality in the space (R n, ‖ · ‖∞). We show that this constant grows to infinity faster than (log n) 1−o(1) when n tends to infinity. To this end, we follow and simplify the approach used by J.M. Ald ..."
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Abstract. We present lower estimates for the best constant appearing in the weak (1, 1) maximal inequality in the space (R n, ‖ · ‖∞). We show that this constant grows to infinity faster than (log n) 1−o(1) when n tends to infinity. To this end, we follow and simplify the approach used by J.M. Aldaz. The new part of the argument relies on Donsker’s theorem identifying the Brownian bridge as the limit object describing the statistical distribution of the coordinates of a point randomly chosen in the unit cube [0,1] n (n large).