Journal of Mathematical Sciences, Vol. 92, No. 3, 1998 STRICTLY STABLE LAWS FOR MULTIVARIATE RESIDUAL LIFETIMES A. A. Balkema (Amsterdam, Holland) and Yong-Cheng Qi (Peking, China) UDC 519.2 The paper is organized as follows: (1) A probabilistic part. We introduce multivariate residual lifetimes and discuss different ways of investigating their asymptotic behavior. We briefly go into the relation with multivariate record sequences. We introduce the concepts of stability and strict stability. (2) An algebraic part. Our problem: to describe the possible connected subgroups of the group of coordinate affine transformations on R a. We give a brief introduction to Lie algebras based on Z. J. Jurek and J. D. Mason. This enables us to give a complete answer to the problem. (3) The description of the strictly multivariate residual lifetime distributions. Introduction Residual lifetimes play an important role in renewal theory, survival analysis, and queueing processes. It is known that the limit laws for residual lifetimes are related to the extreme value limit laws. In this paper, we shall investigate the limit laws in the multivariate setting. The paper is organized as follows: (1) A probabilistic part. We introduce multivariate residual lifetimes and discuss different ways of investigating their asymptotic behavior. We briefly go into the relation with multivariate record sequences. We introduce the concepts of stability and strict stability. (2) An algebraic part. Our problem: to describe the possible connected subgroups of the group of coordinate affine transformations on R d. We give a brief introduction to Lie algebras based on Jurek and Mason [2]. This enables us to give a complete answer to the problem. (3) The description of the strictly multivariate residual lifetime distributions. We may restrict attention to d- dimensional subgroups ~ which do not factorize and do not contain a translation along a vector in [0, oo) d. In appropriate coordinates such a group G consists of all transformations 7 on R e of the form 3,(s t GR, be L, where L is the hyperplane {~ = 0} with ~(s = xt + --- + xd. Assume that d > 1. The noncomposite, strictly stable, residual lifetime distributions on [0, oo) d properly normalized have densities of the form f~ or g~ with a > 0 and L(~)= ~(~+1)'"(~+d-1) (l + ~),'+~-, ' g~'(~)= ~(o,+ l)...(~ +d- I)(I - ~);-' with ~ = xl + "'" + Zd as above, or are uniformly distributed over the d - 1-dimensional simplex generated by the d vectors e'l,- .-, gd of the standard basis of R d 1. Multivariate Residual Lifetimes For a random vector X in R d, we define the residual lifetime X: of X in the point/Y as the vector X -/:7 conditional on X >_/Y. In this setting, it is convenient to work with the tail function R of X defined by R(s = P{X _ s = P{X e [~,~)}. Then the residual lifetime X: is well defined for any point iff, where the tail function is positive and X: has tail function a:(x) = R(:+ ~)IR(p-3, i >_ 6. This relation determines a probability distribution on [0, oo) d. We extend R: to the whole space R a by setting Ry(y-) = R:(ff V O) outside [0, e~) d. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajd6szoboszl6, Hungary, 1997, Part I. 1072-3374/98/9203-3873520.00 9 Plenum Publishing Corporation 3873