Expansions and penultimate distributions of maxima of bivariate normal random vectors

It is well known that the marginal maxima of n$n$ standard normal random vectors with correlation coefficient ρ<1$\rho <1$ are asymptotically independent. In this article, the residual dependence will be captured by asymptotic expansions and certain penultimate distributions including the case where ρ(n)↑1$\rho \left(n\right)\uparrow 1$ at a certain rate.     

A characterization of distributions of independent random vectors by the independence of transformed vectors

If W and Z are independent random vectors and Y₁, Y₂, …, Y_n are the result of a transformation satisfying certain general conditions then W and Z are distributed according to a certain class of densities if and only if for suitable q, (Y₁, …, Y_q) and (Y_q+1, …, Y_n) are independent.     

Laplace approximations for sums of independent random vectors

Summary We consider expressions of the form Z _n =E(exp nF(S _n /n)) where S _n is the sum of n i.i.d. random vectors with values in a Banach space and F is a smooth real valued function. By results of Donsker-Varadhan and Bahadur-Zabell one knows that lim (1/n) log Z _n =sup _x F(x)-h(x) where h is the so-called entropy function. In an earlier paper a more precise evaluation of Z _n is given in the case where there was a unique point maximizing F-h and the curvature at the maximum was nonvanishing. The present paper treats the more delicate problem where these conditions fail to hold.Part of this work has been done when the author was at the Forschungsinstitut für Mathematik in Zürich. I would like to thank the members of the institute and especially Hans Föllmer for the kind hospitality     

Local theorems for sums of independent random vectors

Two multidimensional analogs of the local theorems in L¹ due to Prohorov and Gnedenko, respectively, are derived.     

Laplace approximations for sums of independent random vectors

Summary LetX _i,iN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and a mappingB R. Under some conditions an asymptotic evaluation of is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums under the law transformed by the density exp .     

Estimation of proximity of distributions of sequential sums of independent identically distributed random vectors

Let be a distribution in, let be the n-fold convolution of with itself, and let. It is proved that where depends on certain characteristics of.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 97, pp. 83–87, 1980.In conclusion, the author expresses his gratitude to I. A. Ibragimov for his interest in the paper.     

Laplace approximations for sums of independent random vectors

Let X _i , i∈N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let Φ be a mapping B→R. Under a central limit theorem assumption, an asymptotic evaluation of Z _n = E (exp (n Φ (∑ _{i =1} ⁿ X _i /n))), up to a factor (1 + o(1)), has been gotten in Bolthausen [1]. In this paper, we show that the same asymptotic evaluation can be gotten without the central limit theorem assumption. Received: 19 September 1997 / Revised version:22 April 1999     

Estimating probabilities for sums of independent random vectors

An estimate is made for the probability of occurrence of the value for a sum of independent, identically distributed, random s-dimensional vectors in different families of spheres.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 217–221, February, 1970.     

Extremes of independent chi-square random vectors

We prove that the componentwise maximum of an i.i.d. triangular array of chi-square random vectors converges in distribution, under appropriate assumptions on the dependence within the vectors and after normalization, to the max-stable Hüsler–Reiss distribution. As a by-product we derive a conditional limit result.     

Local limit theorems for sums of independent random vectors

Let p_n(x) and q_n(x) be the densities of the n-fold convolutions of the distributions F and G, respectively. One proves estimates for , expressed in terms of moment characteristics of F — G, under certain restrictions on the densities of F and G. Similar problems are solved for two lattice distributions and for a lattice and a continuous distribution.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 193–212, 1986.     

Limit distributions for the maxima of discrete random variables under monotone normalization

Lithuanian Mathematical Journal - Let X1,X2, . . .,Xn be independent and identical discrete random variables, and let Mn = max(X1, . . .,Xn) denote their maximum. There has been considerable work...     

Convergence of distributions of extremal independent random variables

Kaunas Polytechnic Institute. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 30, No. 2, pp. 219–232, April–June, 1990.     

A new strong invariance principle for sums of independent random vectors

We provide a strong invariance principle for sums of independent, identically distributed random vectors that need not have finite second absolute moments. Various applications are indicated. In particular, we show how one can re-obtain some recent LIL type results from this invariance principle. Bibliography: 16 titles.