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 enormous deviations of the trajectories of random walks, with 'heavy-tailed' (in specific, on a regular basis various, sub- and semiexponential) bounce distributions. It provides a unified and systematic exposition.

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I) If Gi (t)/G(t) → ci as t → ∞, ci 0, i = 1, 2, c1 + c2 > 0, then G1 ∗ G2 (t) ∼ G1 (t) + G2 (t) ∼ (c1 + c2 )G(t). (ii) If G0 (t) ∼ cG(t) as t → ∞, c > 0, then G0 ∈ S. (iii) For any fixed n 2 Gn∗ (t) ∼ nG(t) as t → ∞. 13. It is clear that the asymptotic relation G1 (t) ∼ G2 (t) as t → ∞ defines an equivalence relation on the set of distributions on R. 12(ii) means that the class S is closed with respect to that equivalence. 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).

D. ’s ζ1 , . . 12(ii), one obtains that Gn∨ also belongs to S. d. ’s. This means that ‘large’ values of this sum are mainly due to the presence of a single ‘large’ summand ζi in it. One can easily see that this property is characteristic of subexponentiality. 15. 18 of [113]). 12(ii). 12. (i) First assume that c1 c2 > 0 and that both distributions Gi are concentrated on [0, ∞). 8). ’s. 17) where (see Fig. 1) P1 := P(ζ1 t − ζ2 , ζ2 ∈ [0, M )), P2 := P(ζ2 t − ζ1 , ζ1 ∈ [0, M )), P3 := P(ζ2 t − ζ1 , ζ1 ∈ [M, t − M )), P4 := P(ζ2 M, ζ1 t − M ).

F. 38) is M VI (1/λ) 1/M VI (u/λ) −u e du ∼ VI (1/λ) VI (1/λ) M 1/M e−u du ∼ VI (1/λ). 4(iii) we have VI (1/λ) ∞ M VI (u/λ) −u e du VI (1/λ) VI (1/λ) ∞ ue−u du = o(VI (1/λ)). M Hence (ii) is proved. Assertion (iii) is obvious. ’s is that their regularity character is preserved under convolution. ’s. Let ξ, ξ1 , ξ2 , . . f. 2): F+ (t) ≡ V (t) = t−α L(t). We will denote the class of all such distributions with a fixed α 0 by R(α), and the class of all distributions with regularly varying right tails by R := α 0 R(α).

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