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

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

Example text

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.

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