Janos Galambos

Regularly varying sequences

Abstract : A simple necessary and sufficient condition is developed for a sequence (theta(n)) , n = 0,1,2,.... of positive terms, to satisfy theta(n) = R(n), n > or = 0 , where R(.) is a regularly varying function on (0, infinity). The condition given in the report leads to a Karamata-type exponential representation for theta(n). Various associated difficulties are also discussed. The results are of relevance in connection with limit theorems in various branches of probability theory. (Author)

Order Statistics of Samples from Multivariate Distributions

Abstract Let (X 1j , X 2j , ···, Xmj ), j = 1, 2, ···, n, be a sample of size n on an m-dimensional vector (X 1, X 2, ···, Xm ), m ≥ 2. Let the order statistics of the rth component be denoted by X r,1* ≤ X r,2* ≤ ··· ≤ X r,n *. In this article we investigate the distribution of the vector (X 1,n−i1*, X 2,n–i2*, ···, Xm,n–im *) for (i 1, i 2, ···, im ) not depending on n. The major emphasis is on asymptotic theory and a general formula is given for the asymptotic distribution of the vector above when each ij = 0. Necessary and sufficient condition is also given for the asymptotic independence of the components of the vector investigated. This extends results known for m = 2. In Section 4 examples are given for illustration.

Conditional distributions and the bivariate normal distribution

SummaryAn alaysis of the extent to which conditional distributions of a bivariate vector characterize bivariate normality is given.

Products of random variables : applications to problems of physics and to arithmetical functions

Foundations Limit Theorems Characterization Interacting Particles Arithmetical Functions Miscellaneous Results Bibliography Author Index Subject Index

Some optimal bivariate Bonferroni-type bounds

Let A 1 , A 2 ,..., A 2 and B 1 , B 2 ,..., B m be two sets of events on a probability space. Let X n and Y m be the number of those A j and B s , respectively, that occur. Let S k,t be the (k, t)th binomial moment of the vector (X n , Y m ). We establish optimal bounds on P(X n ≥ 1, Y m ) 1) by means of linear combinations of S 1,1 , S 2,2 , S 1,2 , and S 2,2 . Optimal lower bounds are also determined when only S 1,1 , S 2,1 , and S 1,2 are utilized

The Development of the Mathematical Theory of Extremes in the Past Half Century

This paper is a survey of development of extreme value theory during the last half century from a mathematician’s point of view. The paper considers both general results for classical models (schemes of maximum of independent identically distributed random variables under linear normalization) and extentions of the classical models and the derivations from the classical assumptions also. Different approaches to the characterization theorems for limiting distributions are discribed. The results on the estimation of the rate of convergence in limiting theorems and random sample sizes are given. The paper gives a survey of different approaches to the extreme value theory based on other problems of probability theory.

Bivariate distributions with Weibull conditionals

AbstractВ работе получена пол ная характеризация д вух двумерных распредел ений с вейбулловским и условными распредел ениями в классе всех т ех, которые имеют дважды диффере нцируемые плотности. В одном их н их обе условные плотн ости являются вейбулловс кими, в то время как в другом одн а есть гамма-плотност ь. Метод характеризации, кото рый является общим и применимым ко многим другим случая м условных распределений, полно стью изложен в статье и состоит в ре шении функционально го уравнения, получающе гося из выражения функций совместного распределения в терм инах условных и маргиналь ных плотностей.

Limit laws for mixtures with applications to asymptotic theory of extremes

For a sequence X 1 , X 2 , . . . of random variables, consider the events Aj(x)={Xj>x}, j = 1, 2, ..., where x is an arbitrary real number. Putting v,(x) for the number of Al(x),A2(x),...,A,(x) which occur, the event {v,(x)=0} reduces to {Z,<x}, where Z,=max{X1,X 2 . . . . ,X,}. Here n can be a given integer or a random variable itself. This research has, in fact, started with the aim of unifying techniques for proving limit laws for the extremes when (i) the Xs are independent and n is a random variable, independently distributed of the Xs and when (ii) the Xs are from an infinite sequence of exchangeable random variables and n is a fixed integer. The common property of these two cases is that the distribution of v, (x) can be written in the form

Extreme Value Theory for Applications

Extreme value theory has gone through a rapid development and we can now claim that it has become a mature and significant branch of probability theory. We can also proudly look at the ever increasing number of scientific publications dealing with the applications of extreme value theory. However, as the number of scientific fields, and within each field the number of publications that apply the theory increases, we should also be disturbed by the divergence of the theory and practice. Both theoreticians and applied scientists should listen to each other, seek guidance on what to do and cooperate more and more. This is why it is so delightful to have this opportunity to spend a week together and discuss our subject matter: The theory of extreme values and its applications.

Extensions of Some Univariate Bonferroni-Type Inequalities to Multivariate Setting

Several problems of probability theory lead to the need of estimating the distribution of the number m n = m n (A) of occurrences in a sequence A 1,A 2,…,A n of events. When the estimation of this distribution is in terms of linear combinations of the binomial moments of m n(A), we speak of Bonferroni-type inequalities. That is, let (1.1).