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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Additional resources for Asymptotic analysis of random walks : heavy-tailed distributions
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 ). 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).