4 1. INTRODUCTION very exceptional cases, obtaining a second term is rather hopeless (last statement of section 3.1), while in other cases, obtaining as many terms as one desires can be trivial (penultimate example in section 3.4, dealing with the log-gamma distribution). To conclude this preliminary discussion, and perhaps to motivate further our investigation, we mention that there are many unsolved very basic problems related to linear processes. For instance, for discrete innovations, Davis and Rosenblatt (1991) shows that all but trivial infinite-order linear processes have a continuous marginal distribution but there is still no good criterion to determine if the marginal distribution of the process is absolutely continuous. Even in the most basic cases, the behavior of the marginal distribution is amazingly diﬃcult to analyze. Solomyak’s (1995) breakthrough — a generic result on absolute continuity of the marginal distribution of first-order autoregressive models with Bernoulli innovations — gives a stark reminder on how little is known in general. So this paper can be taken as a contribution to the understanding of those distributions, in the continuous and heavy tail situation. 1.2. Mathematical overview and heuristics. We now outline the mathematical content of the key parts of this paper and provide some intuition behind the main result. In our opinion, the main reason why so few higher-order results are available in the problems we are interested in, is that too much emphasis has been given to an analytical perspective. Our first basic remark is that the convolution operation (F, G) → F G is bilinear and defines a semi-group. Bilinearity is a notion in linear algebra while semi-group is related to the group structure, which in turn suggests representation theory. So our view is that the asymptotic behavior of the convolution semi-group should be analyzed by a linear representation which captures only the tail behavior of the semi-group. In fact we will obtain two linear representations, one which is suitable for writing the expansions, the other one, derived from the first one, and requiring more assumptions, suitable for practical computations. To explain further our view on the subject, let us give a heuristic argument on how to derive higher-order expansions and build the mathematical language to handle these expansions. Heuristic. Let F and G be two distribution functions. Their convolution is F G(t) = F (t − x) dG(x) . The left hand side in this formula is symmetric, while the right hand side, as far as the notation goes, is not. To symmetrize it, we split the integral and integrate by parts to obtain F G(t) = t/2 −∞ F (t − x) dG(x) + t/2 −∞ G(t − x) dF (x) + F (t/2)G(t/2) .

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