We give a necessary and sufficient condition for a d-dimensional Lévy process to be in the matrix normalized domain of attraction of a d-dimensional normal random vector, as t↓0. This transfers to the Lévy case classical results of Feller, Khinchin, Lévy and Hahn and Klass for random walks....

Let [alpha]n and un be the uniform empirical and quantile processes. We investigate the asymptotic distribution of the suprema of [alpha]n(s)/(s(1 - s))1/2±[nu] and un(s)/(s(1 - s))1/2±[nu] with , when the supremum is taken over ranges, depending on n, in the middle of the interval [0, 1],...

Let Gn denote the empirical distribution based on n independent uniform (0, 1) random variables. The asymptotic distribution of the supremum of weighted discrepancies between Gn(u) and u of the forms ||wv(u)Dn(u)|| and ||wv(Gn(u))Dn(u)||, where Dn(u) = Gn(u)-u, wv(u) = (u(1-u))-1+v and 0...

Let X1,n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn,n be the order statistics of n independent random variables with a common distribution function F and let kn be positive numbers such that kn -- [infinity] and . With suitable centering and norming, we investigate the weak...

Stute (1982) and Mason, Shorack and Wellner (1983) have recently completed a thorough study of the limiting behavior of the oscillation of the uniform empirical process. In this paper, the corresponding oscillation behavior of the uniform empirical quantile process is investigated. It is shown...

Let (Ut,Vt) be a bivariate Lévy process, where Vt is a subordinator and Ut is a Lévy process formed by randomly weighting each jump of Vt by an independent random variable Xt having cdf F. We investigate the asymptotic distribution of the self-normalized Lévy process Ut/Vt at 0 and at ∞. We...