and Kilani Ghoudi and Bruno Re millardUniversite du Que bec aÁ Trois-RivieÁ res, Trois-RivieÁ res, Que bec, Canada Let Z 1 , ..., Z n be a random sample of size n 2 from a d-variate continuous distribution function H, and let V i, n stand for the proportion of observations Z j , j{i, such that Z j Z i componentwise. The purpose of this paper is to examine the limiting behavior of the empirical distribution function K n derived from the (dependent) pseudo-observations V i, n . This random quantity is a natural nonparametric estimator of K, the distribution function of the random variable V=H(Z ), whose expectation is an affine transformation of the population version of Kendall's tau in the case d=2. Since the sample version of { is related in the same way to the mean of K n , Genest and Rivest (1993, J. Amer. Statist. Assoc.) suggested that -n[K n (t)&K(t)] be referred to as Kendall's process. Weak regularity conditions on K and H are found under which this centered process is asymptotically Gaussian, and an explicit expression for its limiting covariance function is given. These conditions, which are fairly easy to check, are seen to apply to large classes of multivariate distributions.

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004Alink et al. ( , 2005 in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

×ØÖ ØºWe establish some asymptotic expansions for infinite weighted convolution of distributions having regular varying tails. Various applications to statistics and probability are developed.ÅË ¾¼¼¼ ËÙ Ø Ð ×× ¬ Ø ÓÒ× Primary: 41A60, 60F99. Secondary: 41A80, 44A35, 60E07, 60G50, 60K05, 60K25, 62E17, 62G32.

In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails.Higher-order asymptotics are also obtained when considering a semiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.2000 Mathematics subject classification: primary 62E17; secondary 62E20, 60G70.

The tail behavior of sums of dependent risks was considered by Wüthrich (2003) and by Alink et al. (2004, 2005) in the case where the variables are exchangeable and connected through an Archimedean copula model. It is shown here how their result can be extended to a broader class of dependence structures using multivariate extreme-value theory. An explicit form is given for the asymptotic probability of extremal events, and the behavior of the latter is studied as a function of the indices of regular variation of both the copula and the common distribution of the risks.

We establish some asymptotic expansions for infinite weighted convolutions of distributions having rapidly varying subexponential tails. Examples are presented, some showing that in order to obtain an expansion with two significant terms, one needs to have a general way to calculate higher order expansions, due to possible cancellations of terms. An algebraic methodology is employed to obtain the results.

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

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