Consider the random walk $S_n=\xi_1+...+\xi_n$ with independent and
identically distributed increments and negative mean $\mathbf E\xi=-m<0$. Let
$M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we
present derivation of asymptotics for $\mathbf P(M>x), x\to\infty$ for
long-tailed distributions. This derivation is based on the martingale arguments
and does not require any prior knowledge of the theory of long-tailed
distributions. In addition the same approach allows to obtain asymptotics for
$\mathbf P(M_\tau>x)$, where $M_\tau=\max_{0\le i<\tau}S_i$ and
$\tau=\min\{n\ge 1: S_n\le 0 \}$.Comment: 9 page

Denisov, Denis

Wachtel, Vitali

Electronic Communications in Probability

English

arXiv

OPUS Augsburg

Electron. Commun. Probab. 17 (2012), no. 6, 1–9.DOI: 10.1214/ECP.v17-1757ISSN: 1083-589XELECTRONICCOMMUNICATIONSin PROBABILITYMartingale approach to subexponentialasymptotics for random walks∗Denis Denisov† Vitali Wachtel‡AbstractConsider the random walk Sn = ξ1 + · · · + ξn with independent and identicallydistributed increments and negative mean Eξ = −m < 0. Let M = sup0≤i Si be thesupremum of the random walk. In this note we present derivation of asymptoticsfor P(M > x), x → ∞ for long-tailed distributions. This derivation is based on themartingale arguments and does not require any prior knowledge of the theory oflong-tailed distributions. In addition the same approach allows to obtain asymptoticsfor P(Mτ > x), where Mτ = max0≤i<τ Si and τ = min{n ≥ 1 : Sn ≤ 0}.Keywords: random walk; supremum; cycle maximum; heavy-tailed distribution; stopping time.AMS MSC 2010: Primary 60G70, Secondary 60K30; 60K25.Submitted to ECP on November 30, 2011, final version accepted on January 24, 2012.Supersedes arXiv:1111.6810v1.1 Introduction, statement of results and discussionLet ξ, ξ1, ξ2, . . . be independent random variables with a common distribution func-tion F and negative mean, i.e., Eξ = −a < 0. Let Sn denote the random walk with theincrements ξk, that is,S0 = 0, Sn = ξ1 + ξ2 + · · ·+ ξn, n ≥ 1.It follows from the assumption Eξ < 0 that the total maximum M := supn≥0 Sn is finitealmost surely. The asymptotic behaviour of P(M > x) has been considered by manyauthors. The first results are due to Cramer and Lundberg: if there exists h0 > 0 suchthat Eeh0ξ = 1 and Eξeh0ξ <∞ thenP(M > x) ∼ c0e−h0x as x→∞ (1.1)for some c0 ∈ (0, 1) and, furthermore,P(M > x) ≤ e−h0x for all x > 0. (1.2)The proof of these statements is based on the following observation: The assumptionEeh0ξ = 1 implies that the sequence eh0Sn is a martingale. Applying the Doob inequality∗Supported by the DFG.†School of Mathematics, Cardiff University, Senghennydd Road, CF24 4AG Cardiff, Wales, UK.E-mail: DenisovD@cf.ac.uk‡Mathematical Institute, University of Munich, Theresienstrasse 39, D–80333, Munich, Germany.E-mail: wachtel@mathematik.uni-muenchen.deMartingale approach to subexponential asymptoticswe obtain immediately (1.2). The same martingale property allows one to make anexponential change of measure, which is used in the proof of (1.1).If the distribution of ξ is long-tailed, i.e., Eehξ = ∞ for all h > 0, then one caninvestigate P(M > x) under some additional regularity restrictions on the tail func-tion F (x) := 1 − F (x). One of the most popular regularity assumption is the so-calledsubexponentiality of the distribution tails.Definition 1.1. The distribution function F on R+ is called subexponential if∫ x0F (x− y)dF (y) ∼ 2F (x) as x→∞.The following result is known in the literature as Veraverbecke’s theorem: Let FI bedefined by the tail FI(x) := min(1,∫∞xF (y)dy), x > 0. If FI is subexponential thenP(M > x) ∼ 1aFI(x) as x→∞. (1.3)We next turn to the maximum of the positive excursion of the random walk. Letτ := inf{n ≥ 1 : Sn ≤ 0}andMτ := max0≤n<τSn.If the Cramer-Lundberg condition holds then one can derive the asymptotics for thequantity P(Mτ > x) from that for the total maximum M . This way has been suggestedfirst by Iglehart [10]. Namely, it follows from the Markov property thatP(M > x) = P(Mτ > x) +∫ 0−∞P(M > x− y)P(Sτ ∈ dy,Mτ ≤ x).Thus,P(Mτ > x) = P(M > x)(1−∫ 0−∞P(M > x− y)P(M > x)P(Sτ ∈ dy,Mτ ≤ x)).Noting that (1.1) yieldslimx→∞P(M > x− y)P(M > x)= eh0y for every y < 0,and applying the dominated convergence, we obtain∫ 0−∞P(M > x− y)P(M > x)P(Sτ ∈ dy,Mτ ≤ x) ∼ Eeh0Sτ .As a result we getP(Mτ > x) ∼(1−Eeh0Sτ)P(M > x) ∼(1−Eeh0Sτ)c0e−h0x. (1.4)It turns out that Iglehart’s approach can not be applied to heavy-tailed random walkswithout further restrictions on the distribution of ξ. Here one has to assume that F isstrong subexponential. This class of distribution functions was introduced by Klüppel-berg [11].ECP 17 (2012), paper 6.Page 2/9ecp.ejpecp.orgMartingale approach to subexponential asymptoticsDefinition 1.2. The distribution function F on R belongs to the class S ∗ if∫ x0F (x− y)F (y)dy ∼ 2a+F (x) as x→∞, (1.5)where a+ =∫∞0F (y)dy ∈ (0,∞).Denisov [3] adopted Iglehart’s reduction from Mτ to M to the class of strong subex-ponential distributions: If F ∈ S ∗ thenP(Mτ > x) ∼ EτF (x), x→∞. (1.6)The asymptotics (1.6) were found first by Asmussen [1] for F ∈ S∗ and by Heath,Resnick and Samorodnitsky [9] for regularly varying F . An extension of this result tothe general stopping time can be found in Foss and Zachary [8], and in Foss, Palmowskyand Zachary [7]. These extensions rely on (1.6).The main purpose of the present note is to give alternative proofs of (1.3) and (1.6)using martingale techniques.In order to state our main result we introduce some notation. For any y > 0 letµy := min{n ≥ 0 : Sn > y}.The latter stopping time is naturally connected with the supremum sinceP(M > x) = P(µx <∞).LetF s(x) :=∫ ∞xF (u)duandGc(x) ={F s(x), if x ≥ 0c, if x < 0. (1.7)Define alsoĜc(x) := min{Gc(x), c}. (1.8)Theorem 1.3. Assume that F is long-tailed. For any ε > 0 there exists R > 0 such thatthe stopped sequenceĜa+ε(x− Sn∧µx−R) is a submartingale. (1.9)Assume in addition that F ∈ S∗. For any ε > 0 there exists R > 0 such that the stoppedsequenceĜa−ε(x− Sn∧µx−R) is a supermartingale. (1.10)Having constructed super- and submartingale we can obtain subexponential asymp-totics for M and Mτ by applying the optional stopping theorem.Corollary 1.4. For any long-tailed distribution function F with negative mean,lim infx→∞P(M > x)F s(x)≥ 1a. (1.11)Assume in addition that F ∈ S∗. Then,P(M > x) ∼ 1aF s(x), x→∞.ECP 17 (2012), paper 6.Page 3/9ecp.ejpecp.orgMartingale approach to subexponential asymptoticsTo the best of our knowledge, all existing in the literature proofs of the Veraverbecketheorem are based on representations via geometric sums. More precisely, P(M > x)can be estimated by∑∞n=1(1 − p)pnP(Y1 + Y2 + . . . + Yn > x), where p ∈ (0, 1) and Yiare independent identically distributed random variables with P(Y1 > x) ∼ Fs(x). Inorder to obtain (1.3) from that geometric sum one uses the following two properties ofsubexponential distributions:(a) P((Y1 + Y2 + . . .+ Yn > x)) ∼ nP(Y1 > x) for every fixed k,(b) For every ε > 0 there exists C(ε) <∞ such thatP((Y1 + Y2 + . . .+ Yn > x)) ≤ C(ε)(1 + ε)nP(Y1 > x).A recent elegant proof based on (a) and (b) can be found in [13]. Our proof does not useany property of F besides (1.5).Unfortunately, our method does not allow us to derive (1.3) for the whole class ofsubexponential distributions. The condition FI ∈ S and F ∈ S∗ are close but do notcoincide, see Section 6 in [4]. But we can apply the same construction to Mτ and,as it was shown in [8], the strong subexponentiality is necessary and sufficient forasymptotics (1.6) to hold.Corollary 1.5. Let F ∈ S∗. ThenP(Mτ > x) ∼ EτF (x). (1.12)It is worth mentioning that, in contrast to all previous proofs, our approach to (1.12)is direct, i.e., it does not use any knowledge on the asymptotic behaviour of M .One of the important advantages of the martingale approach is the possibility toobtain non-asymptotic inequalities for P(M > x) and P(Mτ > x). For example, itfollows from (1.9) that for every ε > 0 there exists R > 0 such that (see the proof ofCorollary 1.4)P(M > x) ≥ Fs(x+R)a+ ε, x > 0. (1.13)Using a supermartingale property of Ga−ε we obtain the following upper boundP(M > x) ≤ Fs(x−R′)a− ε, x > R′. (1.14)Of course, in order to apply these inequalities, one has to know how to compute Rand R′ for given values of ε. And we believe that one can do it rather easy for certainsubclasses of S∗, e.g., for regularly varying or Weibull tails.Foss, Korshunov and Zachary [6] have shown that the inequalityP(M > x) ≥ Fs(x)a+ Fs(x), x > 0holds without any restriction on the distribution function F , see Theorem 5.1 in [6].This bound is better than (1.13). It’s proof is based on the fact, that the distribution ofM is the stationary distribution of the Lindley recursion Wn+1 = (Wn + ξn+1)+. Thisproperty of M can be written as follows: Let ξ′ a copy of ξ, which is independent ofM . Then L(M) = L((M + ξ′)). This can be seen as a martingale property: Defineπ(x) := P(M > x). Then the sequence π(x− Sn∧µx) is a martingale.Using (1.10), one gets for all x > R′ the inequalityP(Mτ > x) ≤Fs(x−R′)−EFs(x−R′ − Sτ )a− ε.ECP 17 (2012), paper 6.Page 4/9ecp.ejpecp.orgMartingale approach to subexponential asymptoticsAnd an upper estimate for the difference in the nominator is easy to get:Fs(x−R′)−EFs(x−R′ − Sτ ) = E[∫ x−R′−Sτx−R′F (z)dz]≤ F (x−R′)E[−Sτ ].Applying the Wald identity, we obtainP(Mτ > x) ≤aa− εEτF (x−R′). (1.15)A lower bound is not as obvious. Here we can conclude from (1.9) thatP(Mτ > x) ≥Fs(x+R)−EFs(x+R− Sτ )a+ ε. (1.16)Thus one needs an appropriate estimate for the difference in the nominator.Martingale approach has been used also by Kugler and Wachtel [12] in derivingupper bounds for P(M > x) and P(Mτz > x), where τz := min{k : Sn ≤ −z} underthe assumption that some power moments of ξ are finite. Their strategy is completelydifferent: They truncate the summands ξi in order to construct an exponential super-martingale for the random walk with truncated increments.2 Proofs.2.1 Proof of Theorem 1.3.Fix ε > 0. To prove the submartingale property we need to show thatEĜa+ε(x− y − ξ) ≥ Ĝa+ε(x− y) (2.1)for all y ≤ x−R.Put, for brevity, t := x− y ≥ R. By the definition (1.7),EĜa+ε(t− ξ) = (a+ ε)P(ξ > t− rc) +∫ t−rc−∞F (dz)Fs(t− z)= (a+ ε)F (t− rc) +(∫ t−rc0+∫ 0−∞)F (dz)Fs(t− z),where rc := min{x ≥ 0 : Fs(x) ≤ c}. Integrating the first integral by parts, we obtain∫ t−rc0F (dz)Fs(t− z) = F (0)Fs(t− rc)− F (t− rc)Fs(0) +∫ t−rc0dzF (z)F (t− z).Integrating the second integral by parts, we obtain∫ 0−∞F (dz)Fs(t− z) = F (0)Fs(t)−∫ 0−∞dzF (t− z)F (z).Combining the above inequalities, we getEĜa+ε(t− ξ) = (a+ ε)F (t− rc)− F (t− rc)Fs(0) + F (0)Fs(t− rc)+∫ t−rc0dzF (z)F (t− z) + F (0)Fs(t)−∫ 0−∞dzF (t− z)F (z). (2.2)It is clear that ∫ 0−∞dzF (t− z)F (z) ≤ F (t)∫ 0−∞dzF (z) = a−F (t).ECP 17 (2012), paper 6.Page 5/9ecp.ejpecp.orgMartingale approach to subexponential asymptoticsFurther,F (0)Fs(t− rc) + F (0)Fs(t) = Fs(t) + F (0)∫ tt−rcF (z)dzand ∫ t−rc0dzF (z)F (t− z) =∫ t0dzF (z)F (t− z)−∫ tt−rcdzF (z)F (t− z).Now, put a+ := Fs(0), a− :=∫ 0−∞ dzF (z) and note that a = a− − a+. Consequently,EĜa+ε(t− ξ) ≥ Fs(t) + (a+ ε)F (t− rc)− a+F (t− rc)− a−F (t)+ 2∫ t/20dzF (z)F (t− z) +∫ tt−rcF (z)(F (0)− F (t− z))dz≥ Fs(t) + (−2a+ + ε)F (t− rc) + 2F (t)∫ t/20dzF (z).Now, taking R1 sufficiently large, we can ensure that2∫ t/20F (z)dz ≥ 2a+ −ε2for all t ≥ R1.Furthermore, we can choose R2 so large thatF (t− rc)− F (t)F (t)≤ ε4a+.As a result, for t > max{R1, R2} we haveEĜa+ε(t− ξ) ≥ Fs(t)This proves (1.9).To prove the supermartingale property it sufficient to show thatEGa−ε(x− y − ξ) ≤ Ga−ε(x− y) (2.3)for all y ≤ x−R. Using (2.2) with rc = 0, we obtainEGa−ε(t− ξ) = Ga−ε(t) + (a− ε− a+)F (t)+∫ t0dzF (z)F (t− z)−∫ 0−∞dzF (t− z)F (z).According to the definition of S∗ there exists R1 such that∫ t0dzF (z)F (t− z) ≤ (2a+ + ε/2)F (t)for all t ≥ R1. Furthermore, since F is long-tailed, we havelimt→∞1F (t)∫ 0−∞dzF (t− z)F (z) =∫ 0−∞dzF (z) = a−.Therefore, there exists R2 such that∫ 0−∞dzF (t− z)F (z) ≥ (a− + ε/2)F (t), t ≥ R2.This immediately implies (1.10) with R = max{R1, R2}.ECP 17 (2012), paper 6.Page 6/9ecp.ejpecp.orgMartingale approach to subexponential asymptotics2.2 Proof of Corollary 1.4.Fix ε > 0 and pick R such thatYn = Ĝa+ε(x− Sn∧µx−R)is a submartingale. Then,Fs(x) = Ĝa+ε(x) = EY0 ≤ EY∞= E[Ĝa+ε(x− Sµx−R), µx−R <∞]≤ (a+ ε)P(µx−R <∞).Hence,P(M > x) = P(µx <∞) ≥1a+ εFs(x+R).Letting x to infinity we obtain,lim infx→∞P(M > x)Fs(x)≥ a+ ε.Since ε > 0 is arbitrary the lower bound in (1.11) holds.To prove the corresponding upper bound fix ε > 0 and pick R such that the processYn = Ga−ε(x− Sn∧µx−R) is a supermartingale. Then,Fs(x) = Ga−ε(x) = EY0 ≥ EY∞= (a− ε)P(µx−R <∞, Sµx−R > x)+E[Fs(x− Sµx−R);µx−R <∞, Sµx−R ∈ (x−R, x]]≥ (a− ε)P(µx−R <∞, Sµx−R > x)+ Fs(R)P(µx−R <∞, Sµx−R ∈ (x−R, x]). (2.4)Let r > 0 be a number which we pick later. Then,P(M > x+ r) ≤ P(µx−R <∞, Sµx−R > x)+P(µx−R <∞, Sµx−R ∈ (x−R, x],M > x+ r)≤ P(µx−R <∞, Sµx−R > x)+P(M > r)P(µx−R <∞, Sµx−R ∈ (x−R, x]),where we use the strong Markov property. Now pick sufficiently large r such thatP(M > r) ≤ Fs(R)/(a− ε). Then,P(M > x+ r) ≤ P(µx−R <∞, Sµx−R > x)+Fs(R)a− εP(µx−R <∞, Sµx−R ∈ (x−R, x]).Combining this with (2.4), we getP(M > x+ r) ≤ Fs(x)a− ε.Letting x to infinity we obtain,lim supx→∞P(M > x)Fs(x)≤ 1a− ε.Since ε > 0 is arbitrary the upper bound holds.ECP 17 (2012), paper 6.Page 7/9ecp.ejpecp.orgMartingale approach to subexponential asymptotics2.3 Proof of Corollary 1.5We start with a lower bound. Fix ε > 0 and pick R such that Yn = Ĝa+ε(x−Sn∧µx−R)is a submartingale. Then,Fs(x) = Ĝa+ε(x) = EY0 ≤ EYτ≤ (a+ ε)P(µx−R < τ) +EFs(x− Sτ ).Hence,P(Mτ > x+R) = P(µx+R < τ) ≥1a+ ε(Fs(x)−EFs(x− Sτ )).NowFs(x)−EFs(x− Sτ ) =∫ ∞0P(Sτ ∈ −dt)(Fs(x)− Fs(x+ t))∼ |ESτ |F (x), x→∞. (2.5)By the Wald’s identity |ESτ | = aEτ . Therefore,lim infx→∞P(Mτ > x)F (x)≥ aEτa+ ε.Since ε > 0 is arbitrary we obtain the lower boundlim infx→∞P(Mτ > x)F (x)≥ Eτ.To show the upper bound fix ε > 0 and pick R such that Yn = Ga−ε(x− Sn∧µx−R) is asupermartingale. Then,Fs(x) = Ga−ε(0) = EY0 ≥ EYτ= (a− ε)P(µx−R < τ, Sµx−R > x)+E[Fs(x− Sµx−R);µx−R < τ, Sµx−R ∈ (x−R, x]]+EFs(x− Sτ )≥ (a− ε)P(µx−R < τ, Sµx−R > x) +EFs(x− Sτ )+ Fs(R)P(µx−R < τ, Sµx−R ∈ (x−R, x]).Similarly to the corresponding argument in the proof of Corollary 1.4,P(Mτ > x+ r) ≤ P(µx−R < τ, Sµx−R > x)+P(µx−R < τ, Sµx−R ∈ (x−R, x],Mτ > x+ r)≤ P(µx−R < τ, Sµx−R > x)+P(M > r)P(µx−R < τ, Sµx−R ∈ (x−R, x]),Consequently,P(Mτ > x+ r) ≤1a+ ε(Fs(x)−EFs(x− Sτ )).Now, we can apply (2.5) and obtainlim supx→∞P(Mτ > x)F (x)≤ aEτa− ε.Since ε > 0 is arbitrary we obtain the upper boundlim supx→∞P(Mτ > x)F (x)≤ Eτ.Acknowledgments. This work was partially supported by DFG. This work was donewhile Denis Denisov was visiting the University of Muenchen.ECP 17 (2012), paper 6.Page 8/9ecp.ejpecp.orgMartingale approach to subexponential asymptoticsReferences[1] Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal be-haviour, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8 354–374. MR-1624933[2] Asmussen, S. (2003). Applied probability and queues. 2nd ed., Springer-Verlag, New York.MR-1978607[3] Denisov D. (2005). A note on the asymptotics for the maximum on a random time interval ofa random walk. Markov Processes and related fields 11 165–169. MR-2133349[4] Denisov, D., Foss S. and Korshuov, D. (2004). Queueing Systems 46 No 1-2, 15–33. MR-2072274[5] Feller, W. (1971).An Introduction to Probability Theory and Its Applications. Vol. 2, 2nd ed.,Wiley, New York. MR-0270403[6] Foss, S., Korshunov, D., and Zachary, S. (2011). An Introduction to Heavy-Tailed and Subex-ponetial Distributions. Springer Science+Business Media. MR-2810144[7] Foss, S., Palmowsky, Z. and Zachary, S. (2003). The probability of exceeding a high boundaryon a random time interval for a heavy-tailed random walk. Ann. Appl. Probab. 15 1936–1957.MR-2152249[8] Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walkwith long-tailed increments and negative drift. Ann. Appl. Probab. 13 37–53. MR-1951993[9] Heath, D., Resnick, S. and Samorodnitsky, G. (1997). Patterns of buffer overflow in a classof queues with long memory in the input stream. Ann. Appl. Probab. 7 1021–1057. MR-1484796[10] Iglehart, D.L. (1972). Extreme values in GI/GI/1 queue. Ann. Math. Statist. 43 627–635.MR-0305498[11] Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Probab.25 132–141. MR-0929511[12] Kugler, J. and Wachtel,V. Upper bounds for the maximum of a random walk with negativedrift. ArXiv Preprint: 1107.5400.[13] Zachary, S. (2004). A Note on Veraverbeke’s Theorem . Queueing Systems 46 No 1-2, 9–14.MR-2072273ECP 17 (2012), paper 6.Page 9/9ecp.ejpecp.org

Martingale approach to subexponential asymptotics for random walks

Denis Denisov

Vitali Wachtel

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Consider the random walk S n = ξ 1+,..., + ξ n with independent and identically distributed increments and negative mean Eξ = - m &lt;0. Let M = sup 0 x),x → ∞ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for P(M τ &gt; x), where M τ = max 0&lt;i&lt;τ S i and τ = min{n ≥ 1: S n ≤ 0}.</i

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Martingale approach to subexponential asymptotics for random walks

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