Joint asymptotic distribution of certain path functionals of the reflected process

Let $\tau(x)$ be the first time the reflected process $Y$ of a Levy processes $X$ crosses x>0. The main aim of the paper is to investigate the asymptotic dependence of the path functionals: $Y(t) = X(t) - \inf_{0\leq s\leq t}X(s)$, $M(t,x)=\sup_{0\leq s\leq t}Y(s)-x$ and $Z(x)=Y(\tau(x))-x$. We prove that under Cramer's condition on X(1), the functionals $Y(t)$, $M(t,y)$ and $Z(x+y)$ are asymptotically independent as $\min\{t,y,x\}\to\infty$. We also characterise the law of the limiting overshoot $Z(\infty)$ of the reflected process. If, as $\min\{t,x\}\to\infty$, the quantity $t\te{-\gamma x}$ has a positive limit ($\gamma$ denotes the Cram\'er coefficient), our results together with the theorem of Doney&Maller (2005) imply the existence and the explicit form of the joint weak limit $(Y(\infty),M(\infty),Z(\infty))$.


Introduction
The reflected process Y of a Lévy process X is a strong Markov process on R + .= [0, ∞) equal to X reflected at its running infimum.The reflected process is of great importance in many areas of probability, ranging from the fluctuation theory for Lévy processes (e.g.[2, Ch.VI] and the references therein) to mathematical statistics (e.g.[13,15], CUSUM method of cumulative sum), queueing theory (e.g.[1,14]), mathematical finance (e.g.[9,12], drawdown as risk measure), mathematical genetics (e.g.[11] and references therein) and many more.The aim of this paper is to study the asymptotic dependence and weak limiting behaviour of the functionals of the reflected process Y : (1.1) where t, x ∈ R + .Here τ (x) and Y * (t) denote the first entry time of Y into the interval (x, ∞) and the supremum up to time t of the reflected process respectively, The main result of the paper identifies a condition on the Lévy measure of X, under which the triplet in (1.1) is essentially asymptotically independent in the following sense.A family of random vectors {(U1 z , . . ., U d z )} z∈R l + on a given probability space, where d, l ∈ N, is asymptotically independent if the joint CDF is asymptotically equal to a product of the CDFs of the components: i.e. for any a i ∈ (−∞, ∞], i = 1, . . ., d, it holds (denote a ∧ b .= min{a, b}) as z 1 ∧ . . .∧ z l → ∞.
Under Assumption 1, which is assumed throughout the paper, our main result, Theorem 1, holds.
Theorem 1.The triplet {(Y (t), Z(x + y), M (t, x))} t,x,y∈R + is asymptotically independent and the weak limit Z(x) D −→ Z(∞), as x → ∞, holds, where φ is the Laplace exponent of the increasing ladder-height process of X: The asymptotic independence in Theorem 1 should be contrasted to the intuition that the functionals Z(x + y) and M (t, x) (and hence the triplet in Theorem 1) are unlikely to be asymptotically independent if the Lévy measure of X is heavy-tailed (e.g. if its tail function is regularly varying at infinity).Intuitively, in this case the asymptotic behaviour of the functionals is governed by infrequent but very large jumps that determine the values of Z(x + y) and M (t, x) simultaneously.This is analogous to the behaviour of the path at time t ′ in Figure 1.In fact, in contrast to the heavy-tailed case, the intuitive reason for the asymptotic independence under As. 1 is closely related to the following assertion: the likelihood of a single excursion of Y straddling both the running time t and the first-passage time τ (x + y), as depicted in Figure 1, tends to zero (see remarks following Lemma 3, Section 2.2, for a more detailed intuitive explanation of this phenomenon under As.1).
This fact will be used to establish the asymptotic independence of M (t, x), Z(x + y) and Y (t).
It is not hard to see that in general the functionals M (t, x) and Z(x) are not asymptotically independent as t ∧ x → ∞.We show (see the remark following Lemma 4) that the probability that both random variables occur during the same excursion of the reflected process Y away from zero, as is the case at time t ′ in Figure 1, may not decay to zero under As. 1 However, by Theorem 1, the pair {(M (t, x), Z(x + y))} t,x,y∈R + is asymptotically independent as t ∧ x ∧ y → ∞.Hence, for any α > 1 so are the variables M (t, x) and Z(αx While it may appear intuitively clear that the overshoot Z(x) of a high level x does not occur frequently during the excursion straddling t ′ but before time t ′ (as depicted in Figure 1)-indeed, this excursion still has time to run and reach higher levels, while previous excursions, which have concluded their runs, are more likely to have got to the level x-it does not seems immediately obvious how to make such heuristic arguments precise, particularly since in Theorem 1 the level x is allowed to go to infinity arbitrarily slowly compared to the running time t.One of the contributions of the paper is to establish rigorously the asymptotic independence of Z(x) and Y (t) (cf.Lemma 3 and remarks that follow).
The definition of asymptotic independence of the functionals in (1.1) requires an approximate factorisation of the joint CDF without specifying the rate of divergence or the mutual dependence of t, x, y as they tend to infinity.Hence the asymptotic independence in Theorem 1 does not require the existence of the weak limit of the functionals.It is clear however that the most interesting application of Theorem 1 is precisely in the case when, for each of the functionals, such a limit exist.A result of Doney & Maller [6, Thm 1] implies that M (t, x) converges weakly to a Gumbel distribution if the quantity te −γx tends to a positive constant as t ↑ ∞ (see [7,Ch. 3] for the Gumbel distribution and Appendix A.1, equation (A.3), for a simple derivation of the limit law M (∞) from [6, Thm 1]).
Cramér's condition implies that X tends to −∞ almost surely, and hence, by the classical time reversal argument, the reflected process Y has a stationary distribution Y (∞) equal to the law of the ultimate supremum sup t≥0 X(t).The following corollary of Theorem 1 describes explicitly the various limit laws.
Corollary 2. (i) The weak limit of the random vector (Y (t), Z(x)), as x ∧ t → ∞, exists and the law (Y (∞), Z(∞)) is determined by the joint Laplace transform (ii) Let m .= lim u→∞ φ(u)/u and ν H be the Lévy measure of the Laplace exponent φ with the tail Then the law of the asymptotic overshoot Z(∞) is given by: In particular, Z(∞) is a continuous random variable except possibly at the origin.
A brief description of the proofs and related literature.The main result of this paper is the asymptotic independence in Theorem 1.Its proof, carried out in Section 2.2, is in three steps.
The first and in a certain sense the most important step establishes the asymptotic behaviour of the probability of an event involving the local time at zero of the reflected process.This event contains precisely the paths sketched in Figure 1.In the second step, a splitting property of an extended excursion process of the reflected process, introduced in the classical paper [8], is applied to factorise the probabilities of certain events, related to the ones involving the three functionals but with the running time t replaced by an independent exponential time e(q).The third and final step in the proof applies the factorisations obtained in step two and the asymptotics from the first step to establish the stated asymptotic independence.This is achieved by studying (from first principles) the Laplace inversions of the probabilities arising in step two.
The law of the asymptotic overshoot, given by (1.2) of Theorem 1, is established in two steps in Section 2.3.First, in Proposition 7 we extend the Cramér asymptotics in [3] to the case of the two-sided exit.This step is based on the main result in [3] and a renewal argument from [4], applied in our setting under the Cramér measure.In the second step, the law of Z(x) is expressed in terms of the excursion measure of the reflected process.The limit law is then studied under the excursion measure, using tools such as the result from step one, excursion theory and the asymptotics from [6].
The proof of Corollary 2 is straightforward, once Theorem 1 has been established.Apart from our main result, it involves a number of classical facts from the fluctuation theory of Lévy processes [2].
The details are given in Appendix A.1.

Proof of Theorem 1
The asymptotic independence in Theorem 1 is a consequence of Proposition 6 proved in Section 2.2.
The formula for the law of the asymptotic overshoot follows from Lemma is equal to the right-continuous inverse of L. The ladder-height process We now briefly review elements of Itô's excursion theory that will be used in the proof.We refer to [8] and [2, Ch.IV] for a general treatment and further references.Consider the Poisson point process of excursions away from zero associated to the strong Markov process Y .For each moment where

Definition (2.2) uses the fact L(∞) .
= lim s→∞ L(s) = ∞ P -a.s., which holds by the recurrence of Y .Itô [10] proved that ǫ is a Poisson point process under P .Let n be the intensity (or excursion) measure on (E, G) of ǫ, where G = σ(ǫ(t), t ≥ 0).In Sections 2.2 and 2.3, for any Borel-measurable ) et N q be an independent standard Poisson process with parameter q and consider the process (X, N q ), which is defined on the probability space (Ω × Ω, F ⊗ F N , P × P N ) where F N is the completed filtration generated by {N q (t)} t≥0 and P N the probability law of N q .Let P .= P × P N be the product measure and note that under P the random variable T N q (1), defined by is independent of X and exponentially distributed with mean 1/q.We associate to the Lévy process Under P the process (ǫ, η) is a Poisson point process with values in E × E. To the best of our knowledge, this construction first appeared in [8].We refer to [2, Ch.O.5] for a treatment of Poisson point processes, the compensation formula and the properties of its characteristic measure.(i) The expectation of Then for any δ ∈ (0, ℓ/2) we have (iii) The following limit holds P ( Remarks.(1) Part (iii) in Lemma 3 implies that, as x and t tend to infinity, the probability that the excursion straddling t is the first excursion with height larger than x tends to zero.This fact can be viewed as an intuitive explanation for the asymptotic independence of Z(∞) and Y (∞).Part (iv) of Lemma 3 has analogous interpretation.
(2) The important role played by Lemma 3 in the proof of the asymptotic independence in Theorem 1 lies in the fact that, the limits in parts (iii) and (vi) do not require the point (t, x) in (0, ∞) 2 to tend to infinity along a specific trajectory but only for its norm t ∧ x to increase beyond all bounds.
(3) In contrast to Lemma 3 (iv) the inequality lim sup x∧t→∞ P ( L(τ (x)) < L(t) ≤ L(τ (x + z))) > 0 holds for any fixed z > 0 (cf.remark following the statement of Lemma 4).To show this, recall L(t)/t → ℓ a.s. as t ↑ ∞ (see e.g.proof of Lemma 3 (iii) below) and note that for any small δ > 0 Hence by Lemma 8 and equality (2.38) we find where x ∧ t → ∞ in such a way that te −xγ → λ > 0. Since z > 0, the final inequality clearly holds for δ = 0 and hence by continuity for all δ > 0 sufficiently small.Proof of Lemma 3. Part (i) of the lemma is known.For completeness a short proof, based on the Wiener-Hopf factorisation, is given in the Appendix.
The second limit in part (iv) follows by noting that, for any s ∈ R + and δ 1 ∈ (0, 1/4), the inclusion Since δ 1 can be chosen arbitrarily small, this proves part (iv) and hence the lemma.
Remark.The proof of the asymptotic independence of the triplet (Y (t), Z(x + y), M (t, x)) in Proposition 6 is based on (2.9) and the fact that R 2 (q, x, z) is a linear combination of the probabilities of events, each of which is contained in an event of the form { L(e(q)) = L(τ (x))}, the probability of which tends to zero as t∧x → ∞ (cf.Lemma 3(iii)).It is important to note that the equality in (2.9) cannot be extended to the case z > 0, since the random variables I {Z(x)∈B} and I { L(τ (x+z))< L(e(q))} are clearly functions of the same excursion on the event { L(τ (x + z)) = L(τ (x))} consisting of the paths of Y that cross the levels x and x + z for the first time during the same excursion.In par- ) is in the limit as x → ∞ bounded below by the strictly positive probability of {Z(∞) > z}.This observation invalidates the proof of Lemma 4 if z > 0. Furthermore, it is not difficult to see that in general, for z > 0, the events {M (t, x) ≤ z} and {Z(x) ∈ B} are not asymptotically independent as t ∧ x → ∞.
An analogous argument based on the splitting property of the Poisson point process (ǫ, η) implies that the events {Z(x) ∈ B} and { L(τ (x)) ≤ L(e(q))} are independent.Indeed, let The identities in (2.10) and (2.11) imply the equality in (2.9) in the case z = 0.
Before proving the asymptotic independence of Y (t), Z(x + y) and M (t, x) stated in Proposition 6 below, we need to establish the asymptotic behaviour of certain convolutions that will arise in the proof of Proposition 6.Let T (x) and T (x) denote the first-passage times of X into the intervals (x, ∞) and (−∞, −x) respectively for any x ≥ 0, (2.16) Lemma 5. Let a ∈ [0, ∞) and recall that T (a) is the first-passage time of X over the level a defined in (2.16).Then the following equality holds: (2.17) as y ∧ t → ∞.Furthermore, we have Proof of Lemma 5.The proof of this lemma is based on Lemma 3. Note that for fixed t, y ∈ (0, ∞), the integral in (2.17) can be expressed as an integral over R + (with respect to the measure P (T (a) ∈ ds)) of the integrand s → I [0,t] (s)P ( L(τ (y)) = L(t − s)).Lemma 3 (iii) implies that for any fixed s ∈ R + the integrand tends to zero as y ∧ t → ∞.Therefore (2.17) follows as a consequence of the dominated convergence theorem, since the integrands are uniformly bounded by one and the measure is finite.
To prove equality (2.18), first note that it is equivalent to the statement as y ∧ t → ∞.Since the local time L is non-decreasing, the integrand in (2.19) can be expressed as (2.20) Equality (2.20), Lemma 3 (iv) and the dominated convergence theorem imply that (2.19), and hence (2.18), holds.This completes the proof of the lemma.
Then the following holds as t ∧ y ∧ (x − y) → ∞:

22)
Proof of Proposition 6.We first prove equality (2.21).Note that t ∧ y ∧ (x − y) → ∞ in particular implies t ∧ x → ∞ and t ∧ y → ∞.By (2.8) we have where L −1 denotes the inverse Laplace transform and R 1 (q, x) is defined in Lemma 4. Furthermore, where the second equality holds by Lemma 3 (iii).
To prove (2.21) we therefore need to establish the equality (2.24) Since for every t, Y (t) has the same law as X * (t) = sup 0≤s≤t X(s) , P (∆X(t) = 0) = 1 for all t > 0, where ∆X(t) .= X(t) − X(t−), and {X * (t) > a, ∆X(t) = 0} = {T (a) < t, ∆X(t) = 0}, the following equalities hold: e −qt P (T (a) ∈ dt), (2.25) where as before e(q) is an exponential time with mean 1/q that is independent of X and T (a) is defined in (2.16).Since q → P(Y (e(q)) ∈ A) is by (2.25) the Laplace transform of the positive measure P (T (a) ∈ dt) on R + , the following holds: where the final equality follows by (2.17) in Lemma 5.An analogous argument shows that The definition of R(q, x) in Lemma 4 and the two equalities above imply (2.24) and hence (2.21).
The proof of (2.22) is based on equality (2.27) below, which we now establish.Since by assumption t ∧ y ∧ (x − y) → ∞, for large values of x and y we have 0 ≤ y ≤ x.The definition of r(y, x, q, A, B) in Lemma 4 implies where the final equality follows from Lemma 3 (iii).The identity in (2.9) together with (2.26), identities (2.23), (2.24) and (2.25) and equality (2.18) in Lemma 5 imply: The process X drifts to −∞ as t → ∞ by As. 1, which implies lim t→∞ P (T (a) = t) = 0. Thus, we have the following equality for the set A = (a, ∞): As a consequence the following asymptotic independence holds: Recall that C = (−∞, z] for an arbitrary fixed z ∈ R. In order to prove equality (2.22) note that the following inclusions hold for any y ∈ R + : (recall that τ (x) is defined for x ∈ R + ).These inclusions, together with Lemma 3 (iii), imply that the following equality holds for any family of events E(t, x) ∈ F, t, x ∈ R + , as t ∧ y ∧ (x − y) → ∞: for the fixed z ∈ R + the inequalities 0 ≤ y + z ≤ x hold for all large y and x.In particular (2.27), applied to the complement { L(τ and (2.28) yield the following equalities as t ∧ y ∧ (x − y) → ∞.This concludes the proof of (2.22).

Limiting overshoot.
In this section we prove the formula in (1.2) of Theorem 1, which characterises the law of the limiting overshoot Z(∞).This is achieved in two steps.We first establish Cramér's asymptotics for the exit probabilities of X from a finite interval.In the second step we describe the distribution of the overshoot Z(x), defined in (1.1), in terms of the excursion measure n (see Sections 2.2 for the definition of n) and apply the result from step one to find the relevant asymptotics under the excursion measure, which in turn yield the Laplace transform of the limiting law Z(∞).
For any x ∈ R + , recall that T (x) is given in (2.16) and define the overshoot Denote by f (x) ≃ g(x) as x ↑ ∞ the functions f, g : R + → (0, ∞) satisfying lim x↑∞ f (x) Proposition 7. (i) (Asymptotic two-sided exit probability) For any z > 0 we have where the constant C γ is given in (1.3) and T (z) in (2.16).
Remarks.(i) Let P (γ) x be the Cramér measure on (Ω, F).Its restriction to F(t) is given by Here E x is the expectation under P x and I A is the indicator of A. Under As. 1 it follows that P (γ) x is a probability measure and X − x is a Lévy process under x [X(1) − x] ∈ (0, ∞).(ii) Since the overshoot of X is the same as that of its ladder process, the weak limit under P (γ) of K(x) as x → ∞, needed in the proof of Proposition 7 is be derived from [4,Thm. 1].
(iii) Note that the random variable X( T (z)) under the expectation in (2.29) is well-defined P -a.s., since As. 1 implies that the Lévy process X drifts to −∞ P -a.s.
(iv) The proof of the Proposition 7 is based on two ingredients: the Cramér estimate for Lévy processes [3] and the fact that the overshoot K(x) has a weak limit under P (γ) follows from [4,Thm. 1].The details of the proof are given in Appendix A.2.
For brevity we sometimes write ρ(x) instead of ρ(x, ε).Since the expectation E (γ) [X 1 ] is strictly positive, under P (γ) the reflected process Y is transient and L(∞) is an exponentially distributed random variable, independent of the killed subordinator {( L −1 (t), H(t))} t∈[0, L(∞)) .As a consequence, the excursion process ǫ ′ = {ǫ ′ (t)} t≥0 , defined by the formula in ( Conversely, one may also express n as a ratio of expectations under the measure P .To derive such a representation, for any x > 0, define the random variable K F (x) by (2.32) where the sum runs over all left-end points g of excursion intervals, ǫ g .= ǫ( L(g)), and F : E → R is Borel-measurable and non-negative (note that F ≡ 1 implies K F (x) ≡ 1 P -and P (γ) -almost surely).
. Then the following hold: (iii) For any z ∈ (0, ∞) the following holds as x → ∞:

.36)
A change of variable t = L −1 (u) under the expectation on the right-hand side of (2.36), Fubini's theorem and P The final equality follows from { L −1 (u−) ≤ τ (x)} = {u ≤ L(τ (x))}.Equality in (2.33) under P (γ)   applied to K G (x) and (2.36) now imply the formula in part (ii) of the lemma.
Chebyshev's inequality and part (ii) of the lemma imply e γx n(ε(ρ(z, ε) . The final expression tends to zero as x ↑ ∞ by the dominated convergence theorem and the lemma follows.
We now apply Lemma 9 to establish the asymptotic behaviour of certain integrals against the excursion measure as x → ∞.Lemma 8, in combination with Proposition 10 below, implies the identity in (1.2) thus concluding the proof of Theorem 1.
. = − log E[e −θH(1) I {H(1)<∞} ], for any θ ∈ R + , where I A denotes the indicator of a set A. Analogously, define the local time L of Y at zero, the decreasing ladder-time and ladder-height subordinators L −1 and H with φ the Laplace exponent of H. See [2, Sec.VI.1] for more details on ladder subordinators.Note that the Cramér assumption implies E[X(1)] < 0, making Y (resp.Y ) a recurrent (resp.transient) Markov process on R + .Hence φ(0) > 0 and the stopping time τ (x) is a.s.finite for any x ∈ R + , making H a killed subordinator under P and the overshoot Z(x) a P -almost surely defined random variable.
This schematic figure of a path of Y depicts the values of the three functionals in (1.1) at times t and t ′ , before and after the reflected process crosses the level x + y.It is intuitively clear that, in general, M (t, x), Z(x + y) and Y (t) cannot be independent for fixed t, x, y > 0.
where φ is the Laplace exponent of the decreasing ladder-height process, x) holds for all (t, x) with large t ∧ x.Hence by(2.5)