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Abstract  

We consider an empirical process based upon ratios of selected pairs of spacings, generated by independent samples of arbitrary sizes. As a main result, we show that when both samples are uniformly distributed on (possibly shifted) intervals of equal lengths, this empirical process converges to a mean-centered Brownian bridge of the form B C(u) = B(u)−6Cu(1−u) Σ0 1 B(s)ds, where B(·) denotes a Brownian bridge, and C, a constant. The investigation of the class of Gaussian processes {B C(·): C ∈ ℝ} leads to some unexpected distributional identities such as B 2(·)

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B(·). We discuss this and similar results in an extended framework.

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We describe the limit laws, as t → ∞, of a Bessel process ( R s , st ) of dimension d ∈ (0, 2) penalized by an integrable function of its local time L t at 0, thus extending our previous work of this kind, relative to Brownian motion.

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Let {X n}n∈ℕ be a sequence of i.i.d. random variables in ℤd. Let S k = X 1 + … + X k and Y n(t) be the continuous process on [0, 1] for which Y n(k/n) = S k/n 1/2 for k = 1, … n and which is linearly interpolated elsewhere. The paper gives a generalization of results of ([2]) on the weak limit laws of Y n(t) conditioned to stay away from some small sets. In particular, it is shown that the diffusive limit of the random walk meander on ℤd: d ≧ 2 is the Brownian motion.

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9 20 SHEPP, L. A., A limit law concerning moving averages, Ann. Math. Stat . 35 (1964), 424-428. MR 29 #4091 A limit law concerning moving

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DEHEUVELS, P. and DEVROYE, L., Limit laws of Erdős-Rényi-Shepp type, Ann. Probab. 15 (1987), 1363-1386. MR 88f :60055 Limit laws of Erdős-Rényi-Shepp type

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] Mijnheer , J. 1986 On the rate of convergence to a stable limit law II Litovsk. Mat. Sb. 26 482 – 487 . [10

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. 104 561 574 CHEN, X., On the limit laws of the second order for additive functionals of Harris recurrent Markov chains, Probab. Theory Related

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Artstein, Z. and Hansen, J. C., Convexification in limit laws of random sets in Banach spaces, Ann. Probab. 13 (1985), 307-309. MR 86e :60008 Convexification in limit laws of random

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{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int_0^1 {\tfrac{{ds}}{{r(s)}}\mathop = \limits^{(law)} 2\sup _{s \leqq 1

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