By A. A. Borovkov
This monograph is dedicated to learning the asymptotic behaviour of the possibilities 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
One can easily see that in each equivalence subclass of S under this relation there is always a distribution with an arbitrarily smooth tail G(t). Indeed, let p(t) be an inﬁnitely many times differentiable probability density on R vanishing outside [0, 1]; for instance, one can take p(x) = c exp −1/[x(1 − x)] , x ∈ (0, 1); p(x) = 0, x ∈ (0, 1). Now we will ‘smooth’ the function l(t) := − ln G(t), G ∈ S, by putting l0 (t) := p(t − u)l(u) du, G0 (t) := e−l0 (t) . 8 1 G0 (t) G(t) G(t − 1) →1 G(t) as t → ∞.
Now we will ‘smooth’ the function l(t) := − ln G(t), G ∈ S, by putting l0 (t) := p(t − u)l(u) du, G0 (t) := e−l0 (t) . 8 1 G0 (t) G(t) G(t − 1) →1 G(t) as t → ∞. Therefore G0 is equivalent to the original G. A simpler smoothing procedure that leads to a less smooth asymptotically equivalent tail consists in replacing l(t) by its linear interpolation with nodes at the points (k, l(k)), k being an integer. Thus, up to an additive term o(1), the function l(t) = − ln G(t), G ∈ S, can always be assumed arbitrarily smooth.
D. 1) (cf. 40)). Below we will establish a number of sufﬁcient and necessary conditions that, to some extent, characterize the class S of subexponential distributions. 22 below, p. 29) which is obtained, roughly speaking, through the addition of functions that ‘oscillate’ between the functions from R or between those from Se. 8). 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.