Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk

Transcription

1 Corrected Diffusion Approximations for the Maximum of Heavy-Tailed Random Walk Jose Blanchet and Peter Glynn December, Let (X n : n 1) be a sequence of independent and identically distributed random variables with E (X 1 )=0,andletS =(S n : n 0) be its associated random walk (so that S 0 =0and S n = X X n for n 1). Let us introduce a small location parameter δ > 0 representing the drift of the random walk. That is, let us consider a parametric family of random walks, S δ = S δ n : n 0, defined by S δ n = S n nδ. Our focus here is on developing high accuracy approximations for the distribution of the r.v. M δ =max S δ n 1 n when the increment rv s posses heavy-tails. For our purposes here, we say that X has heavy-tails if for all θ 6= 0, E exp (θ X ) =. (1) In other words, X does not have exponential moments. We say that X has light tails, if E exp (θ X ) < for some θ > 0. The distribution of M δ is of importance in a number of different disciplines. For x>0, {M δ >x} = {τ δ (x) < }, whereτ δ (x) =inf{n 1:Sn δ >x}, sothat computing the tail of M δ is equivalent to computing a level crossing probability for the random walk S δ. Because of this level crossing interpretation, the tail of M δ is of great interest to both the sequential analysis and risk theory communities. In particular, in the setting of insurance risk, P (τ δ (x) < ) is the probability that an insurer will face ruin in finite time (when the insurer starts with initial reserve x and is subjected to iid claims over time); see, for example, Asmussen (2001). The distribution of M δ also arises in the analysis of the single most important model in queueing theory, namely the single-server queue. If the inter-arrival and service times for successive customers are iid with a mean arrival rate less than the mean service rate, then W = (W n : n 0) is a positive recurrent Markov chain 1

2 on [0, ), wherew n is the waiting time (exclusive of service) for customer n. If W is a random variable having the stationary distribution of W, then Kiefer and Wolfowitz (1956) showed that W has the distribution of M δ for an appropriately defined random walk. As a consequence, computing the distribution of M δ is of great interest to queueing theorists. Since W is a positive recurrent Markov chain, the distribution of M δ can be computed as the solution to the equation describing the stationary distribution of W. This linear integral equation is known as Lindley s equation (see Lindley (1952)) and is of Wiener-Hopf type; it is challenging to solve, both analytically and numerically. As a result, approximations are frequently employed instead. One important such approximation holds as δ & 0.. This asymptotic regime corresponds in risk theory to the setting in which the safety loading is small (i.e. the premium charged is close to the typical pay-out for claims) and in queueing theory to the heavy traffic setting in which the server is utilized close to 100% of the time. Thus, this asymptotic regime is of great interest from an application standpoint. Kingman (1963) showed that the approximation P (M δ >x) exp 2δx/σ 2 (2) is valid as δ & 0,whereσ 2 = Var(X 1 ). Becausetherighthandsideof(2)istheexact value of the level crossing probability for the natural Brownian approximation to the random walk S, (2) is often called the diffusion approximation to the distribution of M. However, as with any such approximation, there are applications for which (2) delivers poor results. For the light-tailed case, Siegmund (1979) proposed a so-called corrected diffusion approximation (CDA) that reflects information in the increment distribution beyond the mean and variance. The full asymptotic description in the development initiated by Siegmund was given in the important case of Gaussian random walks by Chang and Peres (1997), and more recently, this complete description in the light tailed case was extended to cover general increment distributions by the authors in Blanchet and Glynn (2004). The strategy followed by Siegmund (1979), Chang and Peres (1997) and Blanchet and Glynn (2004) in the light-tail case makes use of an exponential change-of-measure (see, for example, Siegmund (1985), p. 13.) which yields the convenient representation P (M δ >x)=p (τ δ (x) < ) =exp( θ δ x) E θδ exp θδ S δ τ δ (x) x, (3) where θ δ is assumed to be a positive solution to E exp (θ δ X 1 δθ δ )=1and P θδ generated by the product extension of is df θδ =exp(θ δ x) df. F being the distribution function of X 1. It is clear that, in the heavy-tail context, a direct attack using such change-of-measure ideas is infeasible (since the basic elements 2

3 used to write equation (3) are not well defined in this case.) Thus, in order to develop CDA s for the distribution of M δ one has to approach the problem in a different way. Many problems in applied probability motivate study of models with heavy tails. In risk insurance, for example, statistical evidence suggest that most claims sizes should be modeled as heavy tailed random variables (see, for example, Embrechts, Kluppelberg and Mikosch (1997)). In this context, a popular choice is a Weibull distribution (recall that X Weibull(a, b), fora>0 and b (0, 1] if its support is [0, ) and P (X >x)=exp ax b ), which possesses moments of all orders but is heavy tailed in our sense (for b<1). Queueing theory also gives rise to heavy-tails. Forexample, whenmodelingdatatraffic in communication networks, evidence has been found suggesting that exponential tail features (present in traditional models of data traffic) are not compatible with empirical observations (see Adler, Feldman and Taqqu (1998)). Therefore, developing asymptotic analysis for systems with heavy tail characteristics is an important applied problem. However, as we discussed earlier, developing CDA s in the presence of heavy-tailed increments presents a major mathematical complication as the standard techniques that work well in the presence of light tails (such as exponential change-of-measure) cannot be applied in the heavy tailed context. Also, it often occurs that the nature of the conclusions obtained is completely different for very similar models under these two different tail environments. In the context of risk insurance, for example, it is well known (see Asmussen (2001)) that if the claim distribution is heavy tailed (more precisely, subexponential) the insurance company s bankruptcy is (roughly) driven by a single large claim; in contrast, if the claim size is light tailed then the ruin occurs due to a long sequence of claims that behave according to an (exponentially) twisted version of the underlying claim distribution. For heavy-tailed increments, Hogan (1986) proposed a first order CDA to (2). In particular, assuming that E X 1 5 < and under integrability conditions on the characteristic function of X 1, Hogan showed that as δ & 0 P (M δ >x) exp 2δx/σ 2 (1 2δ/σ) β 1 δ/σ + o (δ). The constant β 1 was computed by Siegmund (1979) as β 1 = 1 6 EX3 1 1 Z 1 2π θ 2 Re log{2(1 g (θ)) /θ2 }dθ. (4) Hogan s strategy consisted, essentially, in applying direct Fourier inversion to the characteristic function of M (δ). His method of proof does not extend directly to higher order correction terms. Our proposed approximation improves upon Hogan s by further reducing the error of the CDA as δ & 0 (in the presence of higher order moments), and by significantly relaxing the integrability assumptions imposed in Hogan s development. In particular, we only require the distribution of X 1 being strongly non-lattice, which means 3

4 that for each ε > 0 inf 1 g (θ) > 0, (5) θ >ε where g (θ) =E exp (iθx 1 ) is the characteristic function of X 1. The form of our approximation is given in the following theorem. Theorem 1 Suppose that E X 1 p < for p>4 and that the distribution of X 1 is strongly non-lattice, then there exist constants (δ) and r (δ) such that P (M δ >x/δ) =exp( (δ) x/δ + r (δ)) + o δ p 4. (6) Moreover, these constants can be computed explicitly in terms of various moments and the characteristic function of X 1 The expressions for the constants (δ) and r (δ) closely resemble their light tail analogues obtained in Blanchet and Glynn (2004). Computing these constants require essentially evaluating a single integral of the form of (4), which makes possible an efficient implementation of approximation (6) in practical situations. In some models with special structure (such as the M/G/1 model in queueing theory or the classical risk model in risk insurance) greatly simplify (avoiding the need for numerical integration). In our view, an important contribution of the paper concerns the method of proof of Theorem 1. The strategy applied to prove Theorem 1 follows a two step procedure. First we use a geometric sum representation for M δ in terms of the ascending ladder heights of S δ (c.f. Kalashnikov (1997), Section 1.3). This representation permits then to formulate the problem so as to take advantage of rates of convergence in the renewal theorem. Second we use a convenient truncated random walk for which light tail techniques (such as (3)) apply. The truncated random walk is shown to be suitably close to S δ via the geometric sum representation and renewal theory arguments, thereby providing the conclusion of the Theorem. The proof of Theorem 1 reveals many useful insights that can be applied to related problems under heavy tail environments. For example, we suggest a plausible corrected diffusion approximation for the probability of ruin within a finite time for insurance companies with heavy tail claims. This problem was declared open by Soren Asmussen (see Asmussen (2001), Ch. 4 remarks). A heuristic approximation was also given by Siegmund (1979) for the light-tailed case (c.f. Siegmund (1985)). It was later adapted to the risk theory context by Asmussen (1982) and has been used successfully (even without a rigorous proof) in many application settings (c.f. Asmussen (1982) and Asmussen and Hojgaard (1999)). To put this problem in perspective note that even in the light-tailed case, the rigorous justification of the finite horizon approximation is known to be hard, see Siegmund (1979). A more compelling argument (although still incomplete) was given later in Theorem of Siegmund (1985). 4

5 An interesting question that remains open arises when the increment distribution has infinite variance (this case is motivated by many models in data traffic andcommunication networks (see Willinger et al (1995)). If Var(X 1 )= but E ( X 1 ) < it is known that there is also a heavy-traffic regime (based on maxima of stable processes) that approximates the distribution of M (see Whitt (2001)). In that case it would be of interest to develop analogous corrected approximations. References [1]Adler,J.,Feldman,R.,andTaqqu,M.(Editors.)A Practical Guide to Heavy Tails: Statistical Techniques and Applications. Birkhauser. [2] Asmussen, S. (1984) Approximations for the Probability of Ruin within Finite Time.Scand.Act.J [3] Asmussen, S. (2001) Ruin Probabilities. World Scientific. [4] Asmussen, S., and Hojgaard, B. (1999) Approximations for Finite Horizon Ruin ProbabilitiesintheRenewalModel.Scand.Act.J.2, [5] Blanchet, J., and Glynn, P. (2004) Corrected Diffusion Approximations for Light- Tailed Random Walk. In preparation. [6] Chang, J. (1992) On Moments of the First Ladder Height of Random Walks with Small Drift. Ann. of App Prob. 2, [7] Chang, J., and Peres, Y. (1997) Ladder Heights, Gaussian Random Walks and the Riemann Zeta Function. Annals of Probability 25, [8] Embrechts, P., Klüppelberg, C., and Mikosch, T. (1999) Modelling extreme events with applications to insurance and finance. Springer-Verlag. [9] Hogan, M. (1986) Comment on Corrected Diffusion Approximations in Certain Random Walk Problems. J. Appl. Probab. 23, [10] Kalashnikov, V. (1997) Geometric Sums: Bounds for Rare Events with Applications. Kluwer. [11] Kiefer, J. and Wolfowitz, J. (1956) On the Characteristics of the General Queueing Process with Applications to Random Walks. Trans. Amer. Math. Soc. 78, [12] Kingman, J. (1963) Ergodic Properties of Continuous Time Markov Processes and Their Discrete Skeletons. Proc. London. Math. soc. 13,

6 [13] Lai, T. (1976) Asymptotic Moments of Random Walks with Applications to Ladder Variables and Renewal Theory. Annals of Probability. 4, [14] Lindey, D. The Theory of a Queue with a Single-Server. Proc. Cambr. Philos. Soc. 48, [15] Siegmund D. (1979) Corrected Diffusion Approximations in Certain Random Walk Problems. Adv. Appl. Prob. 11, [16] Siegmund D. (1985) Sequential Analysis. Springer-Verlag. [17] Whitt, W. (2001) Stochastic-Process Limits. Springer-Verlag. [18] Willinger, W., Taqqu, M., Leland, W., and Wilson, D. (1995) Self-Similarity in High-Speed Packet Traffic: Analysis and Modeling of Ethernet Traffic Measurements. Statistical Science. Vol. 10, [19] Woodroofe, M. (1979) Repeated Likelihood Ratio Tests. Biometrika. 66, [20] Woodroofe, M. (1982) Non-Linear Renewal Theory in Sequential Analysis. Society for Industrial and Applied Mathematics. 6