By A. A. Borovkov
This e-book makes a speciality of the asymptotic habit of the chances of huge deviations of the trajectories of random walks with 'heavy-tailed' (in specific, frequently various, sub- and semiexponential) leap distributions. huge deviation percentages are of significant curiosity in several utilized components, normal examples being smash chances in hazard thought, errors percentages in mathematical information, and buffer-overflow possibilities in queueing conception. The classical huge deviation thought, constructed for distributions decaying exponentially quickly (or even quicker) at infinity, in general makes use of analytical tools. If the short decay fails, that is the case in lots of very important utilized difficulties, then direct probabilistic equipment often turn out to be effective. This monograph offers a unified and systematic exposition of the big deviation idea for heavy-tailed random walks. many of the effects provided within the e-book are showing in a monograph for the 1st time. lots of them have been bought by means of the authors.
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Extra resources for Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions (Encyclopedia of Mathematics and its Applications)
However, these functions usually do not appear in applications, where the assumption of their presence would look rather artiﬁcial. Therefore in what follows we will conﬁne ourselves mostly to considering distributions from the classes R and Se. We will devote less attention to other distributions from the class S. Nevertheless, a number of properties of random walks will be established for the whole broad class S. 1) extends much further (in terms of the number n of summands in the sum of random variables) for distributions from R and Se than for arbitrary distributions from S.
We will often assume that the domain of the functions under consideration includes the right half-axis (0, ∞) where the functions are measurable and locally integrable. 1. ) as t → ∞ if, for any ﬁxed v > 0, L(vt) → 1 as L(t) t → ∞. f. as t → ∞. f. ) as t ↓ 0 is quite similar. f. ) will always refer, unless otherwise stipulated, to a function which is slowly (regularly) varying at inﬁnity. 3) for any ﬁxed v > 0 is a characteristic property of regularly varying functions. f. f. of index zero. 3).
17(ii) enables one to obtain a number of more convenient conditions that are sufﬁcient for subexponentiality. First of all observe that, putting 28 Preliminaries p = 1/2 and M = t/2 in this theorem, we immediately establish the following result from . 19. If G ∈ L and lim supt→∞ G(t)/G(2t) < ∞ then G ∈ S. 17. It extends the condition sufﬁcient for subexponentiality that we established earlier (the regular variation of G(t) at inﬁnity). f. 2)). Following , we will now introduce the following class of functions.
Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions (Encyclopedia of Mathematics and its Applications) by A. A. Borovkov