By Krengel U.

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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 [59] 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 [63], all carry over easily: see [43].

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.