CHAPTER 1 ÒØÖÓ Ù ØÓÒ The primary focus of this paper is to obtain precise understanding of the distribution tail of linear and related stochastic processes based on heavy tail innovations. In doing so, we will develop some new mathematical objects which are tailored to efficiently write and compute asymptotic expansions of these tails. Also, we will derive simple bounds of theoretical importance for the error between the tail and its asymptotic expansion. These tails and their expansions are of interest in a variety of contexts. In the following subsection, we provide some typical examples to illustrate their use. The second subsection of this introduction overviews the new perspective and techniques developed in this paper this will be done at a heuristic level, explaining the intuition and sketching the broad expanse wherein our methods lie. 1.1. Prolegomenom. The first basic problem we will deal with is related to the tail behavior of the marginal distribution of linear processes. To be specific, let C = (ci)i∈Z be a sequence of real constants, and let X = (Xi)i∈Z be a sequence of independent and identically distributed random variables. Let F be the common distribution function of these Xi’s and write F = 1 F for the tail function. Assume that F has a heavy tail, that is, there exists a positive and finite α such that for any positive λ, lim t→∞ F (tλ)/F (t) = λ−α . (1.1.1) Given the sequence C and the distribution function F , we write FC for the distribution function of the series i∈Z ciXi. Set FC = 1 FC. For simplicity, assume in this introduction that the sequence C is nonnegative. Define the series = i∈Z ci α . It is well known that under a mild additional condition, FC CαF at infinity, that is limt→∞ FC(t)/F (t) = Cα. It is then a natural question to investigate higher-order expansions. Under suitable conditions, we will obtain higher-order asymptotic expansions both for FC and its derivatives. In particular all ARMA processes fall within the scope of our result, provided the innovation distribution satisfies certain mild conditions beyond that of heavy tail. 1
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