In this paper, some identities in laws involving ladder processes for random walks and Lévy processes are extended and unified. 2000 Mathematics Subject Classification 60G50, 60G51 (primary), 60G17 (secondary

Alili, L.

Chaumont, L.

Doney, R. A.

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Bull. London Math. Soc. 37 (2005) 141–148 C2005 London Mathematical SocietyDOI: 10.1112/S0024609304003789ON A FLUCTUATION IDENTITY FOR RANDOM WALKSAND LE´VY PROCESSESL. ALILI, L. CHAUMONT and R. A. DONEYAbstractIn this paper, some identities in laws involving ladder processes for random walks and Le´vyprocesses are extended and uniﬁed.1. Fluctuation theory for random walksAn essential component of ﬂuctuation theory in discrete time is the study of thebivariate renewal process formed by the coordinates of the successive maxima ofa random walk S and the corresponding times. This process is called the strictlyor weakly ascending ladder process, depending on whether or not we count thetimes where the maximum is reached with the same value; the descending ladderprocesses are the ascending ladder processes of the dual random walk, −S. Inthe classical era, a breakthrough was achieved by Spitzer, Baxter and others, whointroduced Wiener–Hopf techniques and established several fundamental identitiesthat relate the distributions of the ascending and descending ladder processes tothat of the underlying random walk. These results were originally formulated ina generating function guise, but since then alternative, probabilistic versions havebeen established, and these have led to a reﬁned understanding of the underlyingidentities. In particular, a reformulation of one of these identities by Alili and Doney[2] has led to new results for random walks in [3], [9], [12] and [13], and for Le´vyprocesses in [1] and [6]. In this paper, we establish a further reﬁnement of Aliliand Doney’s identity, for both random walks and Le´vy processes, and give someapplications; in particular, we extend the uniform law given in [13].The formal mathematical setting is ﬁxed as follows. Let S = {Sn : n  0} bea given random walk. That is, S0 = 0 and Sn =∑ni=1 Xi, where {Xi : i  0} isa sequence of independent and identically distributed random variables. The strictascending ladder process (T,H) = {(Tk,Hk) : k  0} is deﬁned as follows:T0 = 0, Tk = inf{n > Tk−1 : Sn > STk−1}, k  1, (1.1)and the heights are given by:Hk = STk , k  0. (1.2)The coordinates of this process are the record heights of the random walk, coupledwith the corresponding indices. It is clear from the Markov property that thisis a bivariate renewal process. The fact that the law of the random walk S isdetermined by the law of the processes (T,H) and (T ∗,H∗), where (T ∗,H∗) isthe weak descending ladder process of the dual walk S∗ = −S, is expressed byReceived 19 August 2003; revised 28 January 2004.2000 Mathematics Subject Classiﬁcation 60G50, 60G51 (primary), 60G17 (secondary).142 l. alili, l. chaumont and r. a. doneythe well-known Wiener–Hopf factorization. This is in turn implicit in the followingidentities, originally due to Sparre-Andersen [14]:(1− E[rT1e−µH1])−1 = exp( ∞∑m=1rmmE[e−µSm I{Sm >0}])(1.3)and (1− E[rT∗1 e−µH∗1 ])−1 = exp( ∞∑m=1rmmE[e−µS∗m I{S∗m0}]), (1.4)which hold for any pair (r, µ) of positive reals. These are the so-called Wiener–Hopffactors. Spitzer [15] and Baxter [4] showed later that relations (1.3) and (1.4) areconsequences of a combinatorial lemma. The aim of the next paragraph is to gathertogether the diﬀerent reformulations of the above identities that have appeared inthe literature.1.1. Variations of Wiener–Hopf factorizationA probabilistic statement that is equivalent to (1.3) is implicit in [10, p. 424,equation 9.3] of Feller. It states that for any n  1 and x > 0, we have1nP (Sn ∈ dx) =n∑k=11kP (Tk = n, Hk ∈ dx). (1.5)In [2], it was shown that for k, n  1 and x > 0, we have1nP (Hk−1 < Sn  Hk, Sn ∈ dx) = 1kP (Tk = n, Hk ∈ dx), (1.6)which is a reﬁnement of relation (1.5). Amongst the results that have been derivedfrom this is a generalized uniform law, discovered by Marchal and given in [13],which we will now recall. We write Gn for the ﬁrst hitting time of the overallmaximum by the random walk S up to time n; that is,Gn = inf{k : Sk = sup0jnSj}, n  1.The special case k = 1 of (1.6) states thatP (H1  Sn ∈ dx) = nP (T1 = n, H1 ∈ dx), x > 0,and Marchal’s result disintegrates this by showing that for any i = 1, . . . , n andx > 0,P (H1  Sn ∈ dx, Gn = i) = P (T1 = n, H1 ∈ dx)=1nP (H1  Sn ∈ dx).(1.7)This says that under the probability P ( · | Sn = x, 0 < Sn  H1), x > 0, the timeGn of the maximum is uniformly distributed over {1, . . . , n}.It is natural to ask whether a more general uniform law can be obtained whenidentity (1.6) is considered for any integer k. More generally, can we unify theidentities (1.5), (1.6) and (1.7)? The answer to this question is provided in thefollowing section; see Theorems 1 and 2.on a fluctuation identity 1431.2. On Feller’s combinatorial lemmaWe start this section by establishing a reﬁnement of Feller’s well-known com-binatorial lemma; see [10, p. 412], and also [7] for the skip-free case. Fix n  1 andlet the sequence s = {si : 0  i  n} be any deterministic chain such that s0 = 0and sn > 0. The associated ladder process (T,H) is deﬁned as in (1.1) and (1.2).We deﬁne the local time at the maximum Ln to beLn = sup{k : Tk  n}. (1.8)By deﬁnition, the cyclic rearrangements θ1(s), . . . , θn−1(s) of s are the shifted chainsdeﬁned for j = 0, 1, . . . , n− 1 byθj(s)i ={si+j − sj , if i  n− j,sj+i−n + sn − sj , if n− j  i  n,i = 0, 1, . . . , n.Lemma 1. There is a unique integer k in {1, . . . , Ln} such thatHLn−k  HLn − sn < HLn−k+1. (1.9)Moreover, k satisﬁes the following properties.(i) The number of cyclical rearrangements of (si) such that n is a ladder time,is k.(ii) The number k is preserved by any cyclical rearrangement of (si).Proof. First, observe that because sn > 0 we have 0  HLn − sn < HLn . Sincek → HLn−k decreases, there exists a unique k ∈ {1, . . . , Ln} satisfying relation(1.9). Now, we may draw a picture in order to see easily that the only cyclicalrearrangements of (si) such that n is a ladder time are those that are obtained bysplitting the chain (si) at a ladder time that is greater than the ﬁrst passage timeabove sup0in si−sn and less than or equal to the ﬁrst hitting time of sup0in si.Note that sup0in si = HLn . From the deﬁnition of k, there are exactly k suchladder times. Statement (ii) is a straightforward consequence of (i).Now let S = {Sn : n  0}, S0 = 0, be a real-valued random walk. As before,let νk be a random variable that is independent of S and uniformly distributed on{0, 1, . . . , k − 1}. SetGkn = TLn−νk ; (1.10)then here is our main result.Theorem 1. For all k and n such that 1  k  n, and for all i = 1, . . . , n, wehaveP (HLn−k  HLn − Sn < HLn−k+1, Sn ∈ dx,Gkn = i) =1kP (Tk = n,Hk ∈ dx).(1.11)Proof. Fix a path S such that HLn−k  HLn − Sn < HLn−k+1. Accordingto Lemma 1(ii), each of the n cyclical rearrangements of S satisﬁes the sameinequalities. Next, according to Lemma 1(i), to each of the n cyclical rearrangementsof S, say S′, there correspond exactly k cyclical rearrangements such that n isa ladder time; these cyclical rearrangements are θTLn−j (S′), j = 0, . . . , k − 1.144 l. alili, l. chaumont and r. a. doneyConversely, the n cyclical rearrangements of each path θTLn−j (S′) (where j =0, . . . , k − 1 are ﬁxed) are the same as those of the initial chain S. Finally,observe that the value Sn is preserved by any cyclical rearrangement, and that{Hk ∈ dx} = {Sn ∈ dx} on the set {Tk = n}.Remark. In fact, Theorem 1 is a consequence of the following pathtransformation, which may also be proved via Lemma 1. Let un be a randomvariable that is independent of S and is uniformly distributed on {0, 1, . . . , n− 1}.Conditionally on {Tk = n}, the chain θun (S) has the same law as the chain Sconditionally on {HLn−k  HLn − Sn < HLn−k+1}. Conversely, conditionally on{HLn−k  HLn −Sn < HLn−k+1}, the chain θGkn (S) has the same law as the chainS, conditionally on {Tk = n}.The following time-reversal property of the ladder process will enable us to stateTheorem 1 in another way; see Theorem 2.Lemma 2. For any integer n  1, the identity in the law stating that{(Tk,Hk) : k  Ln} (d)= {(TLn − TLn−k,HLn −HLn−k) : k  Ln}holds conditionally on Sn.Proof. First we introduce the past maximum process of S: S¯j = supij Si, thenfor k = 0, 1, . . . , Ln, we call e(k) the kth excursion of the reﬂected process S¯ − S:e(k)i = (S¯ − S)Tk +i, 0  i < Tk+1 − Tk,and we denote by e˜(Ln )i the excursion that is in progress at time n:e˜(Ln )i = (S¯ − S)TLn +i, 0  i  n− TLn .By construction, the reﬂected process {(S¯ − S)k : 0  k  n} is the concatenationof e(0), e(1), . . . , e(Ln−1) and e˜(Ln ). Let us denote it by{(S¯ − S)k : 0  k  n} = e(0) ◦ e(1) ◦ . . . ◦ e(Ln−1) ◦ e˜(Ln ).Now, recall that S¯−S is a Markov process. It follows from the Markov property thatif σ is a geometrically distributed random variable that is independent of S, thenconditionally on Lσ, the excursions e(0), e(1), . . . , e(Lσ−1) are i.i.d. and independentof e˜(Lσ ). Hence the processe(0) ◦ e(1) ◦ . . . ◦ e(Lσ−1) ◦ e˜(Lσ ),has the same law as the processe(Lσ−1) ◦ e(Lσ−2) ◦ . . . ◦ e(0) ◦ e˜(Lσ ),and, conditioning on σ = n, we obtain:e(0) ◦ e(1) ◦ . . . ◦ e(Ln−1) ◦ e˜(Ln ) (d)= e(Ln−1) ◦ e(Ln−2) ◦ . . . ◦ e(0) ◦ e˜(Ln ).Then our lemma is a consequence of the above identity.As a consequence, Theorem 1 may be reformulated in the following way.on a fluctuation identity 145Theorem 2. For any integers 1  k  n and for all i = 1, . . . , n, we haveP (Hk−1 < Sn  Hk, Sn ∈ dx, TLn − Tνk = i) =1kP (Tk = n,Hk ∈ dx). (1.12)Remarks.1. Note that all the results of both this section and the next one are also validfor any chain S whose increments are cyclically exchangeable (that is, θj(S) isdistributed as S for each j = 0, 1, . . . , n− 1).2. By summing (1.12) in i, we obtain the Alili–Doney identity (1.6), andMarchal’s identity (1.7) is a consequence of (1.12) for k = 1.1.3. Some consequencesIn the following result, the two ﬁrst assertions are consequences of Theorem 1,whereas the last statement follows from Theorem 2.Corollary 1. For any integers 1  k  n, the following statements hold.(i) Conditionally on the event {HLn−k  Sn < HLn−k+1, Sn = x}, the randomvariable Gkn (deﬁned in (1.10)) is uniformly distributed on {1, . . . , n}.(ii) Summing (1.11) over i yieldskP (HLn−k  HLn − Sn < HLn−k+1, Sn ∈ dx) = nP (Tk = n,Hk ∈ dx).(iii) Conditionally on {Hk−1 < Sn  Hk, Sn ∈ dx}, the random variableTLn − Tνk is uniformly distributed on {1, . . . , n}.Next, we will establish a diﬀerent generalization of Marchal’s uniform law (1.7).In order to simplify the formulas in what follows, let us introduce the time Ny =inf{j  0,Hj  y}, for y > 0. Then observe that {NSn = k} = {Hk−1 < Sn  Hk},for any k. Since T0 ≡ 0, we see that (1.7) says that, conditional on NSn = 1, Gn isuniformly distributed over {T0+1, . . . , n}, and it is natural to ask whether a similarresult holds over {Tk−1 +1, . . . , n} conditional on NSn = k, for any 1  k  n. Forthis, we need the following consequence of (1.7).Lemma 3. For any ﬁxed integer 0 < k  n − 1, and for all pairs of integersi ∈ {k − 1, k, . . . , n− 1} and j ∈ {1, . . . , n} such that i + j  n, we haveP (Tk = n, Hk ∈ dx, Tk−1 = i) = P (NSn = k, Sn ∈ dx, Gn = j + i, Tk−1 = i).(1.13)Proof. It is enough to apply the strong Markov property at the stopping timeTk−1 in the right-hand side of (1.13), and then to use Marchal’s identity (1.7) andthe fact that (T,H) is a renewal process.Remark. We can also prove the above result by a path transform. Indeed,for a ﬁxed pair (i, j) fulﬁlling the above condition, we can establish a one-to-onecorrespondence between paths satisfying {Tk = n, Hk = x, Tk−1 = i} and thosesatisfying {NSn = k, Sn = x, Gn = j + i, Tk−1 = i}. The path transform consistsin leaving unchanged the section of the paths from 0 to i, and applying Marchal’spath transform to the remaining parts of the paths.146 l. alili, l. chaumont and r. a. doneyThe required uniform law follows immediately.Proposition 1. Conditionally on {Sn = x, NSn = k}, the time Gn is uniformlydistributed on {Tk−1+1, Tk−1+2, . . . , n}. That is, for a ﬁxed i ∈ {k−1, k, . . . , n−1},we haveP (Gn = j | NSn = k, Sn = x, Tk−1 = i) =1n− i (1.14)for i < j < n.We also point out that specializing (1.13) to the case j = n− i givesP (Tk−1 = i | NSn = k, Sn = x) = (n− i)knP (Tk−1 = i | Tk = n,Hk = x),and henceE[1n− Tk−1 | NSn = k, Sn = x]=kn. (1.15)This makes an interesting contrast toE[n− Tk−1 | Tk = n,Hk = x]=nk, (1.16)which can also be proved by elementary methods.2. Fluctuation identities for Le´vy processesLet (Xt, t  0), be a real Le´vy process started at 0. We assume that 0 is regularfor (0,∞); that is, P (inf{t : Xt > 0} = 0) = 1. As a consequence, 0 is a regularstate (regular for itself) for the strong Markov process Yt = supst Xs −Xt, calledthe reﬂected process at the maximum; that is, P (inf{t > 0 : Yt = 0} = 0) = 1.Then the process Y admits a local time L at 0, which is deﬁned in the sense ofMarkov processes as in [5, Chapter IV]. Let τ be the right continuous inverse of L:τt = inf{s : Ls > t}, and set Γt = X(τt). Then the process (τ,Γ) is the continuous-time analogue of the discrete ladder process (T,H). More precisely, it is a bivariatesubordinator whose law is characterized as follows. For any couple (α, β) of positivereals, we haveE(e−ατ1−βΓ1)= e−κ(α,β),where the characteristic exponent κ of (τ,Γ) is given byκ(α, β) = k exp(∫∞0∫[0,∞)(e−t − e−αt−βx)t−1P (Xt ∈ dx)). (2.1)The analogue of (1.5) for Le´vy processes was discovered by Bertoin and Doney, andis described in [6]. Now, deﬁne the ﬁrst passage-time process of X byσx = inf{t : Xt > x}, x  0.In [1], the identity between measures on (0,∞)3,t−1P (Lσx ∈ du, Xt ∈ dx) dt = u−1P (τu ∈ dt, Γu ∈ dx) du, (2.2)was proved in the same way that, in discrete time, (1.6) was deduced from (1.3); see[2]. It is then natural to adapt our identities (1.11) and (1.12) to Le´vy processes.In this case, the proof has to be diﬀerent. Indeed, under our regularity assumption,on a fluctuation identity 147the process X has inﬁnitely many ladder times, so that there is no analogue for thecombinatorial Lemma 1. Let us ﬁrst state the following continuous-time equivalentto Lemma 2.Lemma 4. For any t  0, the identity in the law stating that{(τu,Γu) : u  Lt} (d)= {(τLt − τLt−u,ΓLt − ΓLt−u) : u  Lt}holds conditionally on Xt.The proof of this lemma uses classical arguments of excursion theory, and followsthe same scheme as in Lemma 2, so we omit it here.Then the idea of the proof of our result in continuous time is to discretize theprocess X on the time interval [0, 1], in order to obtain a sequence of random walksthat converges almost surely towards X, and such that the corresponding sequenceof ladder processes converges almost surely towards (τ,Γ). Set Mt = supvt Xv.Theorem 3. Let νu be a uniformly distributed random variable on [0, u] thatis independent of X. Then we have the following identity between measures, with(s, t, u, x) ∈ [0, t]× (0,∞)3:P (Lt − LσM t−x ∈ du, Xt ∈ dx, τLt−νu ∈ ds) dt = u−1P (τu ∈ dt, Γu ∈ dx) du ds(2.3)andP (Lσx ∈ du, Xt ∈ dx, τLt − τνu ∈ ds) dt = u−1P (τu ∈ dt, Γu ∈ dx) du ds (2.4)Proof. We ﬁrst give the idea of the proof of (2.4). Note that from the proof of[5, p. 175, Theorem VI.19], x → L(σx) is the right continuous inverse of the processΓ; that isLσx = inf{t : Γt > x}, x  0.Now, for n  1, consider the random walk X(n)k(def)= Xk/n, k  0, and deﬁne itsladder process (T (n),H(n)) as in (1.1) and (1.2), and the ‘local time’ L(n) as in (1.8).Identity (2.4) is then a consequence of Theorem 2 combined with the convergence,as n →∞, of the sequence of three-dimensional processes{(X(n)[nt], n−1T (n)[nt],H(n)[nt]): t  0}towards (X, τ,Γ), in the sense of ﬁnite-dimensional distributions. Then, identity(2.3) follows from identity (2.4) and Lemma 2.The resulting uniform laws are then the same as in Corollary 1 for random walks.Corollary 2. For any ﬁxed t, u, x > 0, the following statements hold.(i) Conditionally on Lt − LσM t−x = u and Xt = x, the random variable τLt−νuis uniformly distributed on [0, t].(ii) Conditionally on Lσx = u and Xt = x, the random variable τLt − τνu isuniformly distributed on [0, t].Remarks.1. Note that τLt is the unique time at which X reaches its overall maximum on[0, t]; that is, τLt = inf{v  t : Xv = sup0lt Xl}. In the above statement, if x = 0,148 on a fluctuation identitythen necessarily u = 0 and Corollary 2 is reduced to the well-known uniform lawgiving the time of the maximum for (cyclically) exchangeable bridges from 0 to 0;see [11] and [8].2. When X is Brownian motion, the uniform laws of Corollary 2 are con-sequences of the path transformation stated in [7, Theorem 7].References1. L. Alili and L. Chaumont, ‘A new ﬂuctuation identity for Le´vy processes and someapplications’, Bernoulli 7 (2001) 557–569 .2. L. Alili and R. A. Doney, ‘Wiener–Hopf factorization revisited and some applications’,Stochastics Stochastics Rep. 66 (1999) 87–102.3. L. Alili and R. A. Doney, ‘Martin boundaries associated with a killed random walk’, Ann.Inst. H. Poincare´ 37 (2001) 313–338.4. G. Baxter, ‘Combinatorial methods in ﬂuctuation theory’, Z. Wahrscheinlichkeitstheorieund Verw. Gebiete 1 (1962/1963) 263–270.5. J. Bertoin, Le´vy processes (Cambridge Univ. Press, 1996).6. J. Bertoin and R. A. Doney, ‘Spitzer’s condition for random walks and Le´vy processes’,Ann. Inst. H. Poincare´ 33 (1997) 167–178.7. J. Bertoin, L. Chaumont and J. Pitman, ‘Path transformations of ﬁrst passage bridges’,Electron. Comm. Probab. 8 (2003) 155–166.8. L. Chaumont, D. G. Hobson and M. Yor, ‘Some consequences of the cyclic exchangeabilityproperty for exponential functionals of Le´vy processes, Se´minaire de Probabilite´s, XXXV,Lecture Notes in Math. 1755 (ed. J. Aze´ma, Springer, Berlin, 2001) 334–347.9. R. A. Doney and P. Marchal, ‘A third arc-sine theorem’, Bull. London Math. Soc. 35(2003) 536–540.10. W. Feller, An introduction to probability theory and its applications, vol. II, 2nd edn(Wiley, New York/London/Sydney, 1971).11. F. B. Knight ‘The uniform law for exchangeable and Le´vy process bridges’, Hommage a`P. A. Meyer et J. Neveu, Aste´risque (1996) 171–188.12. P. Marchal ‘On a new Wiener–Hopf factorization by Alili and Doney’, Se´minaire deProbabilite´s, XXXV, Lecture Notes in Math. 1755 (ed. J. Aze´ma, Springer, Berlin, 2001)416–420.13. P. Marchal ‘Two consequences of a path transform’, Bull. London Math. Soc. 33 (2001)213–220.14. E. Sparre-Andersen, ‘On sums of symmetrically dependent random variables’, Scand.Aktuar. Tidskr. 26 (1953) 123–138.15. F. Spitzer, ‘A combinatorial lemma and its applications to probability theory’, Trans. Amer.Math. Soc. 82 (1956) 323–339.L. AliliDepartment of MathematicsETH-ZentrumHG G 51.2CH-8092, Zu¨richSwitzerlandalilil2002@yahoo.frR. A. DoneyDepartment of MathematicsUniversity of ManchesterOxford RoadManchester M13 9PLUnited Kingdomrad@maths.man.ac.ukL. ChaumontLaboratoire de Probabilite´s etMode`les Ale´atoiresUniversite´ Pierre et Marie Curie4, Place Jussieu - 75252Paris Cedex 05Francechaumont@ccr.jussieu.fr

On a Fluctuation Identity for Random Walks and Lévy Processes

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