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Abstract  

We establish strong limsup theorems related to the law of the iterated logarithm (LIL) for finite dimensional Gaussian random fields by using the second Borel-Cantelli lemma.

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Abstract  

This paper establishes the general moduli of continuity for l -valued Gaussian random fields {X(t):= (X 1(t),X 2(t), h.), t ∈ [0, ∞)N} indexed by the N-dimensional parameter t:= (t 1,…,t N), under the explicit condition yielding that the covariance function of distinct increments of X k(t) for fixed k ≧ 1 is positive or nonpositive.

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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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Abstract  

Strongly inspired by the result due to Carr-Ewald-Xiao that the arithmetic average of geometric Brownian motion is an increasing process in the convex order, we extend this result to integrals of Lévy processes and Gaussian processes. Our method consists in finding an appropriate sheet associated to the original Lévy or Gaussian process, from which the one-dimensional marginals of the integrals will appear to be those of a martingale, thus proving the increase in the convex order property.

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