Subexponential distribution functions in Rd

∞n=0 ∞ pnF ∗n (x), where F ∗n (x) denotes the n-fold convolution of F (x) and where F ∗0 (x) denotes the unit mass at 0. The d.f. W (x )i s called subordinate to F (x) with subordinator {pn}. As in the univariate case in the paper, we shall assume that N satisfies condition (A): N has a generating function P (z )= E(z N ) that is analytic at z =1 . In the present paper, we discuss the relation between the asymptotic behavior of 1 − F (x) and that of 1 − F ∗n (x) and 1 − W (x). It turns out that, as in the univariate case, there are many cases in which 1 − F ∗n (x) asymptotically behaves as n(1 − F (x)) and 1 − W (x) behaves as E(N )(1 − F (x)). To specify the precise kind of asymptotic behavior, we present a form of multivariate subexponentiality. The paper is organized as follows. In Sec. 2, we briefly recall some basic properties and definitions concerning univariate subexponential d.f. In Sec. 3, we introduce and study multivariate subexponential d.f.’s. In Sec. 4, we discuss the relation with regular variation and, in Sec. 5, we provide some extensions. In our main results, we obtain first-order estimates for 1 − F ∗n (x) and 1 − W (x). In a forthcoming paper, we discuss second-order estimates. Without further comment, in the paper, we shall assume that all random vectors X, Y, Z, etc. are positive and have infinite support, i.e., the d.f. satisfies F (0+) = 0 and F (x) < 1, ∀x ∈ R d . We also use the notation F (x )=1 − F (x), and for vectors x and a ,w e setx ◦ = min(xi )a nda ∗ x =( a1x1 ,a 2x2 ,...,a dxd). 2. Univariate Subexponential Distributions In the one-dimensional case, many papers have been devoted to the tail behavior of subordinated d.f.’s. In doing so, the class of subexponential d.f.’s (notation: S) plays an important role. Extending the class S, Chover et al. [6, 7], introduced the class S(γ), where γ ≥ 0. To define these classes, let F (x) denote a d.f. in R such that F (0+) = 0 and F (x) < 1, ∀x ∈ R. Also, let f (s )= E(e −sX ) denote the generating function of X or F (x). The d.f. F (x) belongs to the subexponential class S (notation: F ∈ S) if it satisfies lim