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, frequently various, sub- and semiexponential) bounce distributions. It provides a unified and systematic exposition.

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**Additional info for Asymptotic analysis of random walks : heavy-tailed distributions**

**Example text**

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). One can deﬁne them as measurable functions such that, for all v > 0 from a set of positive Lebesgue measure, there exists the limit lim t→∞ V (vt) =: g(v). g. p. 17 of [32]). The fact that the power function appears in the limit becomes natural from the obvious relation g(v1 v2 ) = lim t→∞ V (v1 v2 t) V (v2 t) × = g(v1 )g(v2 ), V (v2 t) V (t) which is equivalent to the Cauchy functional equation for h(u) := ln g(eu ): h(u1 + u2 ) = h(u1 ) + h(u2 ).

23) ✻ Rpt t t → ∞. ❅ ❅ ❅ t−M ··· M G(pt)G((1 − p)t) ❅ ❅ ❅ (1 − p)t ❅ ❅ ❅ (1−p)t R ❅ ❅ M 0 M pt G(t − y) G(dy) M ❅ ❅ t−M ✲ t ζ1 Fig. 2. 17: for G ∈ S, all three expressions in the plot should be o(G(t)). Proof. To prove both parts of the theorem, it sufﬁces to consider the case G ∈ S+ . 17) (with G1 = G2 = G) for G2∗ (t). 17) (see also Figs. 2). 12(i) that P3 + P4 = o(G(t)) when M = M (t) → ∞ as t → ∞. 17), (1−p)t pt P3 + P4 G(t − y) G(dy) + M G(t − y) G(dy) + G(pt)G((1 − p)t), M so that it sufﬁces to show that P1 ≡ P2 ∼ G(t) as t → ∞.

C. functions. f. c. for any function ψ(t) = o(t). c. for ψ(t) = o t1−α . c. for any ψ. c. function decays (or grows) more slowly than the exponential function (cf. 4(iv)). Now we return to our discussion of subexponential distributions. First we consider the relationship between the classes S and L . 8. 4 hold true for subexponential distributions. This inclusion is strict: not every distribution from the class L is subexponential. 9. 25. 10. c. (and hence for ensuring the ‘subexponential decay’ of the distribution tail, cf.