ARTICLE IN PRESS
Statistics & Probability Letters 77 (2007) 158–165 www.elsevier.com/locate/stapro
The overshoot of a random walk with negative drift Qihe Tang Department of Statistics and Actuarial Science, The University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA Received 20 February 2006; received in revised form 24 May 2006; accepted 29 June 2006 Available online 4 August 2006
Abstract Let fSn ; nX0g be a random walk starting from 0 and drifting to 1, and let tðxÞ be the first time when the random walk crosses a given level xX0. Some asymptotics for the tail probability of the overshoot StðxÞ x, associated with the event ðtðxÞo1Þ, are derived for the cases of heavy-tailed and light-tailed increments. In particular, the formulae obtained fulfill certain uniform requirements. r 2006 Elsevier B.V. All rights reserved. Keywords: Asymptotics; Ladder height; Tail probabilities; The class SðgÞ; Uniformity; Wiener–Hopf type factorization
1. Introduction Let F be the common distribution function of increments of a random walk fSn ; nX0g starting at 0, with F ¼ 1 F satisfying F ðxÞ40 for all x 2 ð1; 1Þ. We assume that F has a finite mean mo0; hence, the random walk Sn drifts to 1 and its ultimate maximum M ¼ maxfS n ; nX0g is finite almost surely. Denote by tðxÞ ¼ inffnX1 : S n 4xg;
xX0,
the first time when the random walk fSn ; nX0g crosses a given level x, with the convention inf f ¼ 1, and denote by AðxÞ ¼ S tðxÞ x the overshoot of the random walk at the level x. As remarked by Chang (1994), the overshoot is among the fundamental objects of study of random walk and renewal theory and therefore it plays an important role in a variety of fields of applied probability. In insurance, the quantity AðxÞ may be interpreted as the deficit at ruin in the renewal model. In the present paper, we are interested in the tail probability of AðxÞ associated with the event ðtðxÞo1Þ. Clearly, PðtðxÞo1Þ40 for all xX0 since F ð0Þ40. We refer the reader to Janson (1986), Asmussen and Klu¨ppelberg (1996), and Klu¨ppelberg et al. (2004) for related discussions. E-mail address:
[email protected]. 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.06.005
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Hereafter, for positive functions ai ðÞ and bi ð; Þ, i ¼ 1; 2, we write a1 ðxÞta2 ðxÞ if lim supx!1 a1 ðxÞ= a2 ðxÞp1, write a1 ðxÞ\a2 ðxÞ if lim inf x!1 a1 ðxÞ=a2 ðxÞX1, and write a1 ðxÞa2 ðxÞ if both limits apply. Moreover, we say that b1 ðx; yÞb2 ðx; yÞ, as x ! 1, holds uniformly for y in some nonempty set D if b1 ðx; yÞ 1 ¼ 0, (1.1) lim sup x!1 y2D b2 ðx; yÞ and we say that b1 ðx; yÞb2 ðx; yÞ, as x ! 1, holds uniformly for yb0 if relation (1.1) holds uniformly for yXf ðxÞ for any positive function f ðxÞ ! 1. We shall apply the work of Veraverbeke (1977) to derive asymptotics for the tail probability PðAðxÞ4y; tðxÞo1Þ under the assumption that the integrated tail distribution of F, defined by Z x 1 F I ðxÞ ¼ R 1 F ðuÞ du; xX0, 0 F ðuÞ du 0 belongs to the class SðgÞ for gX0; see below for its definition. The formula we obtain is Z 1 F ðuÞ du PðAðxÞ4y; tðxÞo1ÞC
(1.2)
xþy
with C40 being explicitly expressed. We establish relation (1.2) in two limit senses: the one is x ! 1 with requirement that relation (1.2) be uniform with respect to y in a relevant infinite interval, and the other is y ! 1 with requirement that it be uniform with respect to x in a relevant infinite interval. As generally acknowledged, the uniformity often significantly merits the asymptotics obtained. 2. The main result We say that a distribution F on ð1; 1Þ or its corresponding random variable X is defective (on the right) if F ð1Þ ¼ PðX o1Þo1. In this case, its right tail is denoted by F ðxÞ R 1¼ F ð1Þ F ðxÞ. For two (possibly defective) distributions F, G, and a real number g, denoted by FbðgÞ ¼ 1 egx F ðdxÞ, if it exists, the moment generating function of F, by F G the convolution of F and G, and by F n the n-fold convolution of F for n ¼ 0; 1; 2; . . . ; with F 0 taking unit mass at 0 and F 1 ¼ F . A distribution F on ð1; 1Þ is said to belong to the class LðgÞ, gX0, if lim
x!1
F ðx uÞ ¼ egu F ðxÞ
for u 2 ð1; 1Þ.
(2.1)
Note that the convergence in (2.1) is automatically uniform on u in any finite interval. Furthermore, a distribution F on ½0; 1Þ is said to belong to the class SðgÞ, gX0, if F 2 LðgÞ and F 2 ðxÞ ¼ 2FbðgÞo1. x!1 F ðxÞ lim
(2.2)
More generally, a (possibly defective) distribution F on ð1; 1Þ is still said to belong to the class SðgÞ, gX0, if the distribution F þ ðxÞ ¼ F ðxÞ=F ð1Þ, xX0, belongs to this class. We remark that if F 2 SðgÞ then g is the right abscissa of convergence of FbðÞ. When g ¼ 0, relation (2.2) describes the famous subexponential class. Recent studies on these classes can be found in Rogozin (2000), Pakes (2004), Shimura and Watanabe (2005), and Tang (2006), among many others. For a proper distribution F with finite mean m, we make a convention that g 1 (2.3) ¼ . m 1 FbðgÞ g¼0
Now we are ready to state the main result of this paper.
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Theorem 2.1. Consider the random walk fS n ; nX0g introduced in Section 1. Suppose F I 2 SðgÞ for gX0 and, in addition, FbðgÞo1 provided g40. (1) As x ! 1, it holds uniformly for yb0 that Z 1 g PðAðxÞ4y; tðxÞo1Þ F ðuÞ du, (2.4) 1 FbðgÞ xþy while when g ¼ 0, the y-region above can be extended to yX0. (2) As y ! 1, relation (2.4) holds uniformly for xb0. Write PðxÞ ðÞ ¼ PðjtðxÞo1Þ for xX0. Under the conditions of Theorem 2.1, it follows from (2.4) and (3.6) that as x ! 1, uniformly for yb0, R1 xþy F ðuÞ du ðxÞ P ðAðxÞ4yÞCðgÞ R 1 (2.5) x F ðuÞ du with CðgÞ ¼ eB ð1 Fbþ ðgÞÞ and B given by (3.3). When g ¼ 0, the coefficient Cð0Þ is equal to 1 and the y-region above can be extended to yX0. Similarly, as y ! 1, relation (2.5) holds uniformly for xb0. By the uniformity of (2.5), it holds for any y40 and any positive function aðxÞ ! 1 that R1 AðxÞ xþaðxÞy F ðuÞ du 4y CðgÞ R 1 PðxÞ ; x ! 1. aðxÞ x F ðuÞ du This enables us to consider the limit distribution, in PðxÞ , of the quantity AðxÞ normalized by a relevant function aðxÞ. For discussions on this topic, we refer the reader to Asmussen and Klu¨ppelberg (1996). 3. Preliminaries Applying Karamata’s representation theorem for regularly varying functions to the class LðgÞ (see Bingham et al., 1987; Klu¨ppelberg, 1989), we know that F 2 LðgÞ if and only if F ðÞ can be written as Z x F ðxÞ ¼ cðxÞ exp bðuÞ du ; xX0, 0
where bðÞ and cðÞ are positive functions such that bðxÞ ! g and cðxÞ ! c40 as x ! 1. By this representation it is easy to verify that for any 40, there is some D40 such that the inequality F ðyÞ pð1 þ Þ expfð1 Þgðy xÞg F ðxÞ
(3.1)
holds whenever yXxXD. For two distributions F 1 and F 2 satisfying F 1 ðxÞcF 2 ðxÞ for some constant c 2 ð0; 1Þ, we know that F 1 belongs to the class SðgÞ or LðgÞ if and only if F 2 belongs to this class. Lemma 3.1. For any g40, the following three assertions are equivalent: R1 (1) F 2 LðgÞ; (2) limx!1 F ðxÞ= x F ðuÞ du ¼ g; (3) F I 2 LðgÞ. Proof. (1) ¼) (2): The proof of this implication is simply an application of the dominated convergence theorem, recalling inequality (3.1), as lim R 1
x!1
x
F ðxÞ 1 ¼ R1 ¼ g. F ðuÞ du 0 limx!1 ðF ðx þ uÞ=F ðxÞÞ du
(2) ¼) (3): With R1 x
F ðxÞ ¼ bðxÞ, F ðuÞ du
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we know that bðxÞ ! g as x ! 1. Integrating this equality with respect to dx over x 2 ð0; y gives that Z 1 Z 1 Z y ln F ðuÞ du ln F ðuÞ du ¼ bðxÞ dx. y
0
That is, the relation Z F I ðyÞ ¼ c exp
0
y
bðxÞ dx 0
holds for some constant c40. This proves that F I 2 LðgÞ. (3) ¼) (1): If we can prove (2), then by the closure of the class LðgÞ under tail equivalence we immediately know that F 2 LðgÞ. To prove (2), for any 40 we derive that Rx ð1=Þ F ðuÞ du 1 F I ðx Þ F I ðxÞ F ðxÞ eg 1 R1 as x ! 1. p R 1 x ¼ ! F I ðxÞ x F ðuÞ du x F ðuÞ du Letting & 0 yields that lim sup R 1 x!1
x
F ðxÞ pg. F ðuÞ du
By the same method we have lim inf R 1 x!1
x
F ðxÞ Xg. F ðuÞ du
This proves (2).
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By Lemma 3.1 and the closure of the class SðgÞ under tail equivalence we immediately obtain the following result: Lemma 3.2. For any g40, F 2 SðgÞ if and only if F I 2 SðgÞ. For the random walk fSn ; nX0g, the well-known Wiener–Hopf-type factorization says that F ¼ F þ þ F F þ F ,
(3.2)
where F þ and F denote the distributions of the upper ladder height and lower ladder height, respectively. In the case when the random walk has a negative drift, i.e., B¼
1 X
n1 PðS n 40Þo1,
(3.3)
n¼1
which is implied by our assumption that the increments distribution F has a finite mean mo0, the distribution F is proper while the distribution F þ is defective with a deficit 1 F þ ð1Þ ¼ eB 40, see Feller (1971, Chapter XII.3) for details. Thus, F þ is understood as F þ ¼ 1 eB F þ . In this case, it was shown by Feller (1971, p. 379) that the distribution R of the maximum M satisfies RðxÞ ¼ eB
1 X
F n þ ðxÞ;
xX0.
(3.4)
n¼0
Veraverbeke (1977) investigated the tail behaviors of the Wiener–Hopf factor F þ and the distribution R. With the help of Lemma 3.2 and relation (2.3), we unify some statements of Theorems 1 and 2 of Veraverbeke (1977) into the following result:
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Lemma 3.3. If F I 2 SðgÞ, gX0, then Z 1 g F þ ðxÞ F ðuÞ du. 1 Fb ðgÞ x If, in addition, FbðgÞo1 provided g40, then for gX0, Z 1 eB g RðxÞ F ðuÞ du. ð1 Fbþ ðgÞÞð1 FbðgÞÞ x
(3.5)
(3.6)
Under the conditions of Lemma 3.3, relations (3.5) and (3.6) indicate that F þ 2 SðgÞ and R 2 SðgÞ, hence b that Fbþ ðgÞo1 and RðgÞo1. It follows from (3.4) that b ¼ RðgÞ
eB . 1 Fbþ ðgÞ
(3.7)
Also, it follows from (3.2) that ð1 Fbþ ðgÞÞð1 Fb ðgÞÞ ¼ 1 FbðgÞ, hence that Fbþ ðgÞo1 if FbðgÞo1.
(3.8)
The following result is from Rogozin and Sgibnev (1999): Lemma 3.4. For some distribution F 2 SðgÞ, gX0, and two other (possibly defective) distributions F 1 and F 2 on ð1; 1Þ such that ki ¼ limx!1 F i ðxÞ=F ðxÞ exists and is finite, i ¼ 1; 2, it holds that F 1 F 2 ðxÞ ¼ k1 Fb2 ðgÞ þ k2 Fb1 ðgÞ. x!1 F ðxÞ lim
Let F 1 and F 2 be two distributions with F 1 2 LðgÞ for gX0. For any yX0 such that F 2 ð½0; yÞ40, it follows from the local uniformity of the convergence in (2.1) that the relation Z y Z y F 1 ðx þ y uÞF 2 ðduÞF 1 ðx þ yÞ egu F 2 ðduÞ (3.9) 0
0
holds as x ! 1. Now we make the statement somewhat stronger. Lemma 3.5. For some distribution F 2 SðgÞ, gX0, and two other (possibly defective) distributions F 1 and F 2 on ½0; 1Þ such that ki ¼ limx!1 F i ðxÞ=F ðxÞ exists and is positive, i ¼ 1; 2, relation (3.9) holds uniformly for y 2 LðF 2 Þ ¼ fu : F 2 ðuÞ40g. Proof. For any large M40, by the local uniformity of the convergence in (2.1), we have R y F ðx þ y uÞF ðduÞ 2 0 1 Ry sup lim 1 ¼ 0. x!1 y2LðF Þ;ypM F 1 ðx þ yÞ egu F 2 ðduÞ 2 0 Thus, it suffices to prove that R y F ðx þ y uÞF ðduÞ 2 0 1 Ry lim lim sup sup 1 ¼ 0, M!1 x!1 y4M F 1 ðx þ yÞ egu F 2 ðduÞ 0 which amounts to the conjunction of the inequalities Ry F 1 ðx þ y uÞF 2 ðduÞ Ry lim sup lim sup sup 0 p1 x!1 y4M F 1 ðx þ yÞ 0 egu F 2 ðduÞ M!1
(3.10)
and Ry lim inf lim inf inf M!1
x!1 y4M
F 1 ðx þ y uÞF 2 ðduÞ Ry X1. F 1 ðx þ yÞ 0 egu F 2 ðduÞ 0
(3.11)
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Let X 1 and X 2 be two independent and nonnegative random variables distributed by F 1 and F 2 , respectively. Clearly, Z y F 1 ðx þ y uÞF 2 ðduÞ ¼ PðX 1 þ X 2 4x þ y; 0pX 2 pyÞ 0
pPðX 1 þ X 2 4x þ yÞ PðX 1 þ X 2 4x þ y; 0pX 1 pxÞ Z x ¼ F 1 F 2 ðx þ yÞ F 2 ðx þ y uÞF 1 ðduÞ. 0
By this, Lemma 3.4, and Fatou’s lemma, we obtain Ry F 1 ðx þ y uÞF 2 ðduÞ Ry lim sup lim sup sup 0 x!1 y4M F 1 ðx þ yÞ 0 egu F 2 ðduÞ M!1 Rx F 1 F 2 ðx þ yÞ 0 F 2 ðx þ y uÞF 1 ðduÞ p lim sup lim sup sup RM x!1 y4M M!1 F 1 ðx þ yÞ 0 egu F 2 ðduÞ Z x 1 F 1 F 2 ðxÞ F 2 ðx þ y uÞ lim ¼ lim inf lim inf inf F 1 ðduÞ x!1 y4M 0 M!1 F 1 ðx þ yÞ Fb2 ðgÞ x!1 F 1 ðxÞ Z 1 1 k2 b F 2 ðx uÞ b p F 2 ðgÞ þ F 1 ðgÞ lim inf F 1 ðduÞ x!1 k1 F 1 ðxÞ Fb2 ðgÞ 0 ¼ 1. This prove (3.10). As for (3.11), again applying Fatou’s lemma we have Ry F 1 ðx þ y uÞF 2 ðduÞ Ry lim inf lim inf inf 0 M!1 x!1 y4M F 1 ðx þ yÞ egu F 2 ðduÞ 0 Z M 1 F 1 ðx þ y uÞ lim inf lim inf inf X F 2 ðduÞ F 1 ðx þ yÞ Fb2 ðgÞ M!1 x!1 y4M 0 Z 1 1 F 1 ðx uÞ lim inf X F 2 ðduÞ x!1 b F 1 ðxÞ F 2 ðgÞ 0 ¼ 1. This ends the proof of Lemma 3.5.
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4. Proof of Theorem 2.1 4.1. Proof of item 1 By copying the proof of Lemma 3.1 of Tang (2003) we can obtain that for each xX0 and yX0, Z x PðAðxÞ4y; tðxÞo1Þ ¼ eB F þ ðx þ y uÞRðduÞ.
(4.1)
0
First, we aim at a lower asymptotic bound. By (4.1) and the local uniformity of the convergence in (2.1), for any M40, as x ! 1, it holds uniformly for yX0 that Z M PðAðxÞ4y; tðxÞo1Þ\eB F þ ðx þ y uÞRðduÞ 0 Z M egu RðduÞ. eB F þ ðx þ yÞ 0
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Since M40 is arbitrary, as x ! 1, it holds uniformly for yX0 that b PðAðxÞ4y; tðxÞo1Þ\eB F þ ðx þ yÞRðgÞ.
(4.2)
Substituting (3.5) and (3.7) into (4.2) and using (3.8), we obtain that uniformly for yX0, Z 1 g eB PðAðxÞ4y; tðxÞo1Þ\eB F ðuÞ du 1 Fb ðgÞ xþy 1 Fbþ ðgÞ Z 1 g ¼ F ðuÞ du. 1 FbðgÞ xþy
ð4:3Þ
Next, we aim at upper asymptotic bounds for the cases g ¼ 0 and gX0, respectively. When g ¼ 0, by equality (4.1) it holds for all xX0 and yX0 that Z xþy PðAðxÞ4y; tðxÞo1ÞpeB F þ ðx þ y uÞRðduÞ ¼ Rðx þ yÞ, 0
where the last step can be verified by (3.4). Thus by (3.6), we have that as x ! 1, uniformly for yX0, Z 1 1 F ðuÞ du. (4.4) PðAðxÞ4y; tðxÞo1Þt m xþy Combining (4.3) with (4.4) we prove the assertion for g ¼ 0. For the general case of gX0, we rewrite (4.1) as PðAðxÞ4y; tðxÞo1Þ Z xþy Z B B F þ ðx þ y uÞRðduÞ e ¼e 0
xþy
F þ ðx þ y uÞRðduÞ
x
¼ I 1 I 2. The same as above, I 1 ¼ Rðx þ yÞ. For I 2 , by interchanging the order of integrals and after some subtle analysis we have Z xþy Z 1 I 2 ¼ eB F þ ðdvÞRðduÞ x xþyu Z y B B ¼ e ð1 e ÞRðx þ yÞ þ F þ ðyÞRðxÞ þ Rðx þ y uÞF þ ðduÞ . 0
It follows that PðAðxÞ4y; tðxÞo1ÞpeB Rðx þ yÞ eB
Z
y
Rðx þ y uÞF þ ðduÞ.
(4.5)
0
Applying Lemma 3.5, as x ! 1, it holds uniformly for yb0 that Z y Rðx þ y uÞF þ ðduÞRðx þ yÞFbþ ðgÞ. 0
Substituting this into (4.5) and using (3.6) and (3.8), we obtain that as x ! 1, uniformly for yb0, PðAðxÞ4y; tðxÞo1ÞteB Rðx þ yÞ eB Rðx þ yÞFbþ ðgÞ Z 1 g F ðuÞ du. ð1 FbðgÞÞ xþy Combining (4.3) with (4.6), we prove the assertion for gX0.
ð4:6Þ
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4.2. Proof of item 2 Applying Lemma 3.5 to (4.1), we immediately obtain that as y ! 1, the relation Z x PðAðxÞ4y; tðxÞo1ÞeB F þ ðx þ yÞ egu RðduÞ 0
holds uniformly for xX0. Hence, it holds uniformly for xb0 that b PðAðxÞ4y; tðxÞo1ÞeB F þ ðx þ yÞRðgÞ;
y ! 1.
(4.7)
Substituting (3.5) and (3.7) into (4.7) and using (3.8), we obtain the announced result. Acknowledgment The author wishes to thank the referee for his/her careful reading of the previous version of this paper. At the stage of proofreading the author was informed that Lemma 3.1 of the present paper had earlier been obtained by Su et al. (2004). References Asmussen, S., Klu¨ppelberg, C., 1996. Large deviations results for subexponential tails, with applications to insurance risk. Stochastic Process. Appl. 64 (1), 103–125. Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. Cambridge University Press, Cambridge. Chang, J.T., 1994. Inequalities for the overshoot. Ann. Appl. Probab. 4 (4), 1223–1233. Feller, W., 1971. An Introduction to Probability Theory and its Applications, vol. II, second ed. Wiley, New York, London, Sydney. Janson, S., 1986. Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift. Adv. in Appl. Probab. 18 (4), 865–879. Klu¨ppelberg, C., 1989. Subexponential distributions and characterizations of related classes. Probab. Theory Related Fields 82 (2), 259–269. Klu¨ppelberg, C., Kyprianou, A.E., Maller, R.A., 2004. Ruin probabilities and overshoots for general Le´vy insurance risk processes. Ann. Appl. Probab. 14 (4), 1766–1801. Pakes, A.G., 2004. Convolution equivalence and infinite divisibility. J. Appl. Probab. 41 (2), 407–424. Rogozin, B.A., 2000. On the constant in the definition of subexponential distributions. Theory Probab. Appl. 44 (2), 409–412 Translated from Rogozin, B.A., 1999. Teor. Veroyatnost. i Primenen. 44(2), 455–458 (in Russian). Rogozin, B.A., Sgibnev, M.S., 1999. Banach algebras of measures on the line with given asymptotics of distributions at infinity. Siberian Math. J. 40 (3), 565–576 Translated from Rogozin, B.A., Sgibnev, M.S., 1999. Sibirsk. Mat. Zh. 40(3), 660–672 (in Russian). Shimura, T., Watanabe, T., 2005. Infinite divisibility and generalized subexponentiality. Bernoulli 11 (3), 445–469. Su, C., Chen, J., Hu, Z., 2004. Some discussions on the class LðgÞ. J. Math. Sci. 122 (4), 3416–3425. Tang, Q., 2003. A note on the severity of ruin in the renewal model with claims of dominated variation. Bull. Korean Math. Soc. 40 (4), 663–669. Tang, Q., 2006. On convolution equivalence with applications. Bernoulli 12 (3), 535–549. Veraverbeke, N., 1977. Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stochastic Processes Appl. 5 (1), 27–37.
