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, on a regular basis various, sub- and semiexponential) bounce distributions. It offers a unified and systematic exposition.
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Additional resources for Asymptotic analysis of random walks : heavy-tailed distributions
Note that in the literature the notation S is normally used for the class of subexponential distributions on [0, ∞), whereas the general case is either ignored or just mentioned in passing as a possible extension obtained as above. ’s are always positive. In the context of general random walks, such a restrictive assumption is no longer natural. e. ’s) may well cause confusion, especially when the concept of subexponentiality is encountered for the ﬁrst time. 12(iii) below). v. v. 10 on p. 19).
Since l(t) → ∞, we can assume without loss of generality that γ0 + 1 γ0 γ := . 38) with s0 = t0 , denote by h(t) the continuous piecewise linear function with nodes at the points (sn , l(sn )), n 0: h(t) := l(sn ) + l(sn+1 ) − l(sn ) (t − sn ), z(sn ) t ∈ [sn , sn+1 ]. 39), we have thus deﬁned the function h(t) on the entire half-line [t0 , ∞) (its deﬁnition on the left of the point t0 is inessential for our purposes). 42) since γ0 < γ < 1. 45) t − z(sn ) γ0 γ0 l(sn ) γ0 h(t) = z(sn ) sn t − z(sn ) h(t) γ0 1 − z(sn )/sn t γh(t) , t since l(sn ) l(t0 ).
34) as follows: ∞ ε/λ V (1/λ) = λ ∞ e 2 V (1/λ) du = L(1/λ) λ −u −α L(u/λ) u ε ∞ + ε . 4(iii), for all sufﬁciently small λ one has L(u/λ)/L(1/λ) < u for u 2). Therefore ∞ ∼ ε/λ V (1/λ) λ ∞ u−α e−u du. 37) ε Now observe that, as λ ↓ 0, εV (ε/λ) λ L(ε/λ) V (1/λ) = ε1−α → ε1−α . 32). 1 Regularly varying functions and their main properties obtain for α = 1 and M > 1 that ∞ ∞ −λt ψ(λ) = e VI (t) de−λt dVI (t) = − 0 0 1/M ∞ = VI (u/λ)e −u du = 0 + 0 ∞ M + 1/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).