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Xt c, and hence b(t) → c∆α as t → ∞. 10) with t = τ n to reverse the argument. From this, and the analogous statements which hold for limsup and liminf, known results about L´evy processes such as strong laws and laws of the iterated logarithm can easily be deduced. But there is a vast literature on the asymptotic behaviour of random walks, and by no means all the results it contains have been extended to the setting of L´evy processes. Using Theorem 13 we can show, for example, that the classical results of Kesten in  about strong limit points of random walks, and results about the limsup behaviour of Sn /nα and |Sn |/nα and hence about ﬁrst passage times outside power-law type boundaries in , all carry over easily: see .
2) 0 The second result we need is a slight extension of one we’ve seen before, in Chapter 2; here and throughout, we write a(x) ≈ b(x) to signify that ∃ absolute constants 0 < C1 < C2 < ∞ with C1 ≤ a(x)/b(x) ≤ C2 for all x ∈ (0, ∞) and write C for a generic positive absolute constant. Lemma 4. If U is the renewal function of any subordinator having killing rate k, drift δ, and L´evy measure µ, and x µ(y)dy, A(x) = δ + 0 then U (x) ≈ x . 18) therein. For subordinators it appears as Proposition 1, p.
Proof. 11), ˆ + (Yˆ − ) < and furthermore that I + (respectively I − ) is ﬁnite if and only if E B − ˆ+ ˆ ∞ (respectively E B (Y ) < ∞). As previously mentioned, Proposition 7 is valid with lim replaced by lim inf or lim sup . The results then follow from Theorem 14. 1 Introduction In the last ten years or so there have been several new developments in connection with the Wiener–Hopf equations for L´evy processes, and in this chapter I will describe some of them, and try to indicate how each of them is tailored to speciﬁc applications.