By A. A. Borovkov

ISBN-10: 052188117X

ISBN-13: 9780521881173

This monograph is dedicated to learning the asymptotic behaviour of the chances of huge deviations of the trajectories of random walks, with 'heavy-tailed' (in specific, usually various, sub- and semiexponential) bounce distributions. It offers a unified and systematic exposition.

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**Extra resources for Asymptotic analysis of random walks : heavy-tailed distributions**

**Sample text**

12) x0 where d(x) = ln c(ex ) → d ∈ R and δ(x) = ε(ex ) → 0 as x → ∞, x0 = ln t0 . 12) for the function h(x). First of all note that h(x) (like L(t)) is a locally bounded function. 2, for a large enough x0 and all x x0 sup |h(x + y) − h(x)| < 1. 9) the bound |h(x) − h(x0 )| x − x0 + 1. Further, the local boundedness and measurability of the function h mean that it is locally integrable on [x0 , ∞) and therefore can be represented for x x0 as x0 +1 h(x) = 1 (h(x)−h(x+y)) dy+ h(y) dy+ x0 x 0 (h(y+1)−h(y)) dy.

Put h(x) := ln L(ex ). 8) as x → ∞. To prove the theorem, we have to show that this convergence is uniform in u ∈ [u1 , u2 ] for any ﬁxed ui ∈ R. 8) is uniform on the interval [0, 1]. 9) we have |h(x + u) − h(x)| (u2 − u1 + 1) sup |h(x + y) − h(x)|, u ∈ [u1 , u2 ]. y∈[0,1] For a given ε ∈ (0, 1) and any x > 0 put Ix := [x, x + 2], ∗ I0,x Ix∗ := {u ∈ Ix : |h(u) − h(x)| := {u ∈ I0 : |h(x + u) − h(x)| ε/2}, ε/2}. Ix∗ ∗ are measurable and differ from each other only It is clear that the sets and I0,x ∗ ∗ ), where μ is the Lebesgue measure.

27) Then G0 ∈ S. Proof. 19 and the above remarks. 17(ii) for p = 1/2. 5 on p. 17). Clearly, G0 ∈ L (cf. 4(i)). 21) for G0 (t), p = 1/2 and the chosen M = M (t). 20) also holds for G0 (t). 17(i)). 21) for G0 (t). The theorem is proved. 3 Further sufﬁcient conditions for distributions to belong to S. 17 essentially gives necessary and sufﬁcient conditions for a distribution to belong to S. 21, which, as we will see from what follows, are quite narrow. To construct broader (in a certain sense) sufﬁcient conditions, we will introduce the class of so-called semiexponential distributions.

### Asymptotic analysis of random walks : heavy-tailed distributions by A. A. Borovkov

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