A renewal theorem and supremum of a perturbed random walk

We study tails of the supremum of a perturbed random walk under regime which was not yet considered in the literature. Our approach is based on a new renewal theorem, which is of independent interest. We obtain first and second order asymptotics of the solution to renewal equation under weak assumptions and we apply these results to obtain first and second order asymptotics of the tail of the supremum of a perturbed random walk.


Introduction
1.1. Renewal theorems. Almost every renewal quantity may be described as the solution f to an integral equation where µ is a probability measure and ψ is a locally bounded function. When functions f , ψ and measure µ are supported on R + , then f is given by (2) f (x) = where H = ∞ n=0 µ n , provided lim n→∞ ψ * µ n = 0. Such equations appear frequently in different problems. In particular, they are closely related to stochastic fixed point equations, [6].
In the general case when ψ and µ are defined on R and µ has a strictly positive mean, the second equality in (2) does not hold but quite likely, due to properties of ψ and µ, (0,x] ψ(x − y)H(dy) may become the main term in the asymptotics of f (x) as x → ∞. The classical Key Renewal Theorem (KRT) gives the asymptotics of f at infinity when ψ is directly Riemann integrable but in applications one encounters equations (1) where corresponding ψ is not even in L 1 and so there is a need of a more general theory.
In this paper we study behavior of the integrals (0,x] L e x−z H(dz) and R L e x−z H(dz), as x → ∞, where L is a slowly varying function with the property that L(e x ) is not integrable on R + . The asymptotics of (3) does not follow from the classical KRT nor from known relatives (see e.g. [8,Section 6.2.3]), because the latter results are obtained under additional assumption that the integrand function is (ultimately) monotone or is asymptotically equivalent to a monotonic function. In our case, x → L (e x ) may exhibit infinite oscillations, so in general it is not asymptotically equivalent to a monotonic function. Slow variation of L plays here the role of the regularity condition. For 0 ≤ x 0 < x, let If the integral exists, L(x 0 , x) is again slowly varying as a function of x. Under very mild assumptions on L, we show that (Theorem 3.1) for any x 0 > 0, where m is the mean of µ. Here and henceforth, f (x) ∼ g(x) means that f (x)/g(x) → 1 as x → ∞. The proof of Theorem 3.1 is surprisingly simple and although we use particular properties of slowly varying functions, the scheme behind it may be adopted to other situations. Under some further assumptions we are able to reduce the second quantity in (3) to the first one and to obtain the second order asymptotics (Theorem 3.3). Imposing some more regularity on µ, we show that , as x → ∞.

Supremum of a perturbed random walk. Consider
where the sequence (A n , B n ) n≥1 is i.i.d. and A 1 ≥ 0 a.s. The renewal equation (1) arises naturally from the study of the right tail of R. When A 1 and B 1 are positive a.s. then taking the logarithm of both sides of (5), M = log R is the supremum of the so-called perturbed random walk (PRW) where X 1 = log A 1 , ξ 1 = log B 1 . In a more general case considered here it is natural to call A 1 ·. . .·A n−1 ·B n a perturbed multiplicative random walk. Random variable R, if exists, satisfies the following fixed-point stochastic equation P(R > x) converges to zero when x tends to infinity and it is a natural question to describe the rate at which it happens. We do it here under specific assumptions and the result clearly applies to M . PRW is a natural extension of the random walk with applications to queuing theory, insurance risk as well as to telecommunication networks [8], [14], [15], [16]. Supremum of the process with regenerative increments can be represented as M for an appropriate PRW and the size of the largest box in the Bernoulli sieve as R, [8]. Therefore, supremum of PRW is a natural object to study. Supremum of PRW, M , inherits characteristics related to max n {X 1 + · · · + X n } and to the extreme behavior of the perturbations ξ n . Tail behavior of M is studied in three main regimes in [14], one of them being the Cramer case which requires existence of some exponential moments of X 1 and ξ 1 . Since our framework is a little bit more general, we shall explain it in terms of A 1 = e X1 , B 1 = e ξ1 . In the Cramer case the tail behavior of R may be determined by A or B alone, or by both of them. The first case happens when the tail of B is regularly varying with index −α < 0, EA α < 1 and EA α+ε < ∞ for some ε > 0.
On the other hand, if EA α = 1, EB α + < ∞, ρ = EA a log A < ∞ and the distribution of log A given A > 0 is non-arithmetic, then ([6, Theorem 5.2]) and it is A that plays the main role here. The Cramer case when both A and B contribute significantly to the tail has not been considered yet for R although it is quite natural to do it, see the example described in Subsection 1.3.
We assume that there exists α > 0 such that EA α = 1, B has a regularly varying right tail with index − α, EB α + = ∞ (see Section 4 for the rest of assumptions) and we describe the right tail of R (Theorem 4.2). If P(B > x) = x −α L(x) for x > 0, where L is a slowly varying function, then and so the tail is essentially bigger than that of B, since L(0, x)/L(x) → ∞ as x → ∞; see (10). Appearance of the function L is probably the most interesting phenomenon here. To obtain (7) we use a renewal theorems mentioned in the previous section. A somehow related problem was dealt with in [12]. Finally, assuming some more regularity on the distribution of A, we obtain the second order asymptotics in (7), that is, see Theorem 4.2 (ii). Note that if L is asymptotically bounded away from zero, then we have We hope that (7) holds in a more general setting: where ψ : R → R is a random Lipschitz mapping such that for any and EC α i < ∞. But then calculations become much more technical and it is still a work in progress.

Extremes of perturbed random walk.
There is a somehow related problem, where contributions to asymptotics of some statistic may come from one of two ingredients alone or from both of them. Let (ξ n , η n ) n be a sequence of i.i.d. two-dimensional random vectors with generic copy (ξ, η). Consider the maximum of PRW, The aim is to study convergence in distribution of a n M n for some suitable chosen deterministic sequence (a n ) n≥1 . There are essentially three distinct cases. In the first case Eη 2 < ∞, S n dominates the perturbation and the limit of a n M n coincides with the limit of a n max 1≤k≤n {S k−1 }. In the second one, the tail P(η > x) is regularly varying with index γ ∈ (−2, 0), perturbation η n dominates the random walk and the limit coincides with the limit of a n max 1≤k≤n {η k }. For above see [7,Theorem 3]. In the most interesting, third case, that is, if P(η > x) ∼ cx −2 for some c > 0, both random walk and the perturbation have comparable contributions, see [18,9] along with generalization to functional limit theorems. Further developments have been made recently in [10], where the assumption Eξ 2 k < ∞ is dispensed with.

Preliminaries
The class of such functions will be denoted R(ρ). If f ∈ R(0) then f is called a slowly varying function. The class of slowly varying functions plays a fundamental part in the Karamata's theory of regular variability, since if f ∈ R(ρ), then f (x) = x ρ L(x) for some L ∈ R(0). Below, we introduce some basic properties of the class R(0) that, later on, will be essential.
If L ∈ R(0) is bounded away from 0 and ∞ on every compact subset of [0, ∞), then for any δ > 0 there exists A = A(δ) > 1 such that (Potter's Theorem, see e.g [5], Appendix B) Assume that L ∈ R(0) is locally bounded on (X, ∞) for some X > 0. Then, for α > 0 one has and this result remains true also for α = 0 in the sense that since the convergence in (8) is locally uniform outside zero [3, Theorem 1.5.2]. Moreover, since implies that L is slowly varying. In the theory of regular variation, L is called the de Haan function.
2.2. Renewal theory. Let (Z k ) k≥1 be the sequence of independent copies of a random variable Z with EZ > 0. We write S n = Z 1 + . . . + Z n for n ∈ N and S 0 = 0. The measure defined by We say that the distribution of Z is arithmetic if its support is contained in dZ for some d > 0 and non-arithmetic, otherwise. Equivalently, the distribution of Z is arithmetic if and only if there exists 0 = t ∈ R such that f Z (t) = 1, where f Z is the characteristic function of the distribution of Z. The law of Z is strongly non-lattice if the Cramer condition is satisfied, that is, lim sup |t|→∞ |f Z (t)| < 1.
A fundamental result of renewal theory is the Blackwell theorem [4]: if the distribution of Z is nonarithmetic, then for any t > 0, Note that in the non-arithmetic case, since H ((x, for some positive α, β and any h > 0. Under additional assumptions we know more about the asymptotic behavior of H (see [17]). If for some r > 0 one has P(Z ≤ x) = o(e rx ) as x → −∞, then there exists r 1 > 0 such that Exact asymptotics of H ((−∞, x]) as x → −∞ in the presence of α > 0 such that Ee −αZ = 1 are given in [13].
Finally, if Z has finite second moment and for some r > 0, P(Z > x) = o(e −rx ) as x → ∞ and the distribution of Z is strongly non-lattice, then for some r 1 > 0 (see [17])

Renewal Theorems
There are many variants of this theorem, when f is not necessarily L 1 -see [8, Section 6.2.3]. Such results are usually obtained by additional requirement that f is (ultimately) monotone or f is asymptotically equivalent to a monotone function.
Here we obtain a renewal result that is essentially stronger: an asymptotic of for a slowly varying function L, Theorem 3.1. Such a function may exhibit infinite oscillations, so in general it is not asymptotically equivalent to a monotonic function.
Theorem 3.1 Assume that 0 < EZ < ∞ and the law of Z is non-arithmetic. Let L be a slowly varying function, which is locally bounded on [1, ∞). For any x 0 ≥ 0 such that L(x 0 , x) is finite and L(x 0 , x) → ∞ as x → ∞, one has

Remark 3.2
The assumption that the law of Z is non-arithmetic is not crucial here. The same result holds if one assumes that the law of Z is arithmetic and the proof of such result requires only small modifications necessitated by the use of the Blackwell theorem in the arithmetic case.
Under stronger assumptions, particularly assuming that x → x −α L(x) is a monotonic function, we may prove second order asymptotics. Theorem 3.3 Assume that 0 < EZ < ∞, Ee εZ < ∞ for some ε > 0, the law of Z is strongly non-lattice and P(Z ≤ x) = o(e rx ) as x → −∞ for some r > 0. Assume further that there is a random variable B, a slowly varying function L and a constant α > 0 such that Proofs of both renewal theorems are postponed to Section 5. We note only that for any slowly varying function there exist α > 0 and another slowly varying function L 0 such that L(x) ∼ L 0 (x) and x −α L 0 (x) is the tail of a probability distribution. In this sense, the assumption of existence B in Theorem 3.3 is not very restrictive. 4. Tails of the supremum of perturbed random walk 4.1. Notation and assumptions. Throughout the paper, log stands for the natural logarithm. We are going to write a + for max{a, 0}. For any n ≥ 1 we write Π n = A 1 · . . . · A n and Π 0 = 1. Our standing assumptions are: (A-1) P(A ≥ 0) = 1, the law of log A given A > 0 is non-arithmetic, As an easy consequence of (9) we obtain Proposition 4.1 Suppose that (B-1) is satisfied. Then EB α + 1 B≤x = α L(x)−L(x) ∼ α L(x) and for any r > 0, .
In this chapter the previous results in the renewal theory will be applied to the random variable Z with the law defined by 4.2. Tails of perturbed multiplicative random walk. In this section we study the asymptotics of P(R > x), where R is defined in (5). Under (A-2) and (B-1) with β < α, we have EA β < 1 and EB β < ∞. Since R + has finite moments up to α. If one assumes additionally that (i) If (18) holds, then (ii) If EA α+ε < ∞ for some ε > 0 and additionally the distribution of Z defined by (17) is strongly non-lattice, then

Remark 4.3
(1) We say that the law µ is spread-out if there exists n ∈ N such that n-th convolution µ * n has a non-zero absolutely continuous part. Notice that if the law of log A is spread-out then the law of Z is spread-out and so it is strongly non-lattice. If the law of A has a non-trivial absolutely continuous component then the same holds for log A implying that the law of Z is strongly non-lattice and we have (19).
Proof. Let f (x) = e αx P(R > e x ) and ψ(x) = e αx (P(R > e x ) − P(AR > e x )). Let g : R n → R be a Borel function. If (Z k ) k is an i.i.d. sequence with the law (17), then Eg(Z 1 , . . . , Z n ) = EΠ α n g(log A 1 , . . . , log A n ), where (A k ) k are i.i.d. In particular, Then, (c.f. (1)) Iterating (21) we obtain where S n = Z 1 + . . . + Z n , S 0 = 0. Clearly, if the law of log A given A > 0 is non-arithmetic under P, then the law of Z is non-arithmetic as well.
By (20), the random walk (S n ) n≥1 has positive drift, thus S n → ∞ with probability 1 as n → ∞. Moreover, since R has finite moments up to α, by Markov inequality we have f (x) ≤ ER α−ε e εx for ε ∈ (0, α). Thus, where H is the renewal measure of (S n ) n≥1 .
In our case ψ is not dRi (it is not even in L 1 ), so the Key Renewal Theorem is not applicable. Instead, we consider ψ B (x) = e αx P(B > e x ) = L(e x ) and define ψ 0 = ψ − ψ B . First we will show that R ψ 0 (x − z)dH(z) is convergent as x → ∞ to a finite limit. Therefore, R ψ B (x − z)dH(z) will constitute the main part (see Theorems 3.1 and 3.3). Indeed, ψ 0 (x) = −e αx P(min{AR, B} > e x ) and where D = log min{AR, B}. Using Fubini's theorem and changing the variable t = z − x + D, we obtain By (12), we may take the limit as x → ∞ inside the integral. Thus, by the Blackwell Theorem we get For the main part, we have Let us concentrate now on the first order asymptotics, point (i). We will show that For any δ > 0, the integrand is bounded by ce −δz for some c > 1 by Potter bounds. Combining this with (13) and Lebesgue's Dominated Convergence Theorem we conclude that [Asymptotics of H ((−∞, x]) which is more precise than (13) is available here (see [13]): again by the Lebesgue Dominated Convergence Theorem. The first part of the assertion will follow from Theorem 3.1. Indeed, we already know that the expectation of Z is strictly positive and finite. Moreover, the law of Z is non-arithmetic. Thus, For the purpose of second order asymptotics (ii), we additionally assume that EA α+ε is finite and that the law of Z is strongly non-lattice. Observe that E exp(εZ) = EA α+ε < ∞ and thus, the assumptions of Theorem 3.3 are satisfied. Thus, , as x → ∞.
So far we have shown that where is the error term coming from the integral of ψ 0 . However, L may be decreasing to 0 (e.g. L(t) ∼ 1/ log(t)) and we want to be more precise here. We will show that for some δ > 0, .

Proofs
Proof of Theorem 3.1. Using the definition of a slowly varying function, it is easy to see that the integral over (0, since the convergence in (8) and, by the Blackwell theorem, the right hand side above converges. Thus, it is enough to concentrate on the integral over (x 0 , x − x 0 ]. Let us fix n ∈ N and ε > 0 and observe that Further, by Potter bounds ([3, Theorem 1.5.6 (i)]), if z ∈ (x 0 + k/n, x 0 + (k + 1)/n], for any δ > 0 and sufficiently large x and x 0 we have L(e x−z ) L(e x−x0−k/n ) ≤ (1 + ε)e δ/n ≤ (1 + ε) 2 and similarly for the lower bound. Moreover, let x 0 be such that for any k ≥ 0, ≤ H x 0 + k n , x 0 + k + 1 n ≤ (1 + ε) 1 nEZ .
Altogether, above considerations yield We already know that the first term is asymptotically equivalent to L(exp(x))H((−∞, 0]) (see (23)). The second term equals L(exp(x)) ∞ 0 exp(−αz)H(dz) and the integral is convergent, thus it is of the same order as the first one.
The main contribution comes from the third term, which is equal to where R(z) = H((−∞, z]) − z/EZ. Since ∞ 0 L(exp(x − z))dz = L(0, exp(x)), we have and after integrating by parts and changing the variable t = z − x + log B, It remains to show that K 2 (x) = O(L(e x )).
We have by Proposition 4.1.