Asymptotic tail behavior of phase–type scale mixture ... - arXiv

Asymptotic tail behavior of phase–type scale mixture distributions Leonardo Rojas-Nandayapa

arXiv:1502.01811v1 [math.PR] 6 Feb 2015

School of Mathematics and Physics The University of Queensland Brisbane, QLD, Australia [email protected]

Wangyue Xie School of Mathematics and Physics The University of Queensland Brisbane, QLD, Australia [email protected]

Abstract We consider phase–type scale mixture distributions which correspond to distributions of random variables obtained as the product of two random variables: a phase–type random variable Y and a nonnegative but otherwise arbitrary random variable S called the scaling random variable. We prove that the distribution of X := S · Y is heavy–tailed iff the distribution of the scaling random variable S has unbounded support. We also investigate maximum domains of attraction. For the Fréchet domain of attraction, we prove a converse of Breiman’s lemma which provides the exact asymptotics of the tail probability of X. Sufficient conditions are given for a phase–type scale mixture distribution to be in the Gumbel domain of attraction. We explore subexponentiality for some important cases of phase–type scale mixtures. Our results are complemented with several examples. Keywords: phase–type; exponential; Erlang; scale mixtures; infinite mixtures; heavy–tailed; subexponential; maximum domain of attraction; products.

1

Introduction

In this paper we consider the general class of nonnegative distributions defined by the Mellin–Stieltjes convolution (Bingham et al., 1987) of two nonnegative distributions G and H, given by Z ∞ F (x) = G(x/s)dH(s), x ≥ 0. (1.1) 0

1

A distribution of the form (1.1) will be called a phase–type scale mixture if G is a phase–type distribution and H is a proper nonnegative distribution that we shall call the scaling distribution. Recall that phase–type distributions correspond to distributions of absorption times of Markov jump processes with one absorbing state and a finite number of transient states—the exponential, Erlang and Hyperexponential distributions are examples of phase–type distributions. A phase–type scale mixture distribution may be seen as the distribution of random variable X of the form X := S · Y where S ∼ H and Y ∼ G. We shall call S a scaling random variable. Using a conditional argument we obtain that (X|S = s) ∼ Gs , where Gs (x) := G(x/s) corresponds to the distribution of the (scaled) random variable s·Y ; we shall call Gs a scaled phase–type distribution. Therefore, the distribution F can be thought as a mixture of the scaled phase–type distributions in {Gs : s > 0} with respect to the scaling distribution H. In this paper we focus on the analysis of the asymptotic behavior of phase–type scale mixture distributions. We wish to investigate conditions for phase–type scale mixture distributions to be heavy–tailed and determine their maximum domains of attraction. Our motivation is to approximate heavy–tailed distributions via phase–type scale mixtures. Heavy–tailed distributions are of key importance in various branches of applied probability for modeling random phenomena exhibiting extreme behavior (cf. Embrechts et al., 1997). Nevertheless, the analysis of many important stochastic models with heavy–tailed inputs is difficult because heavy–tailed distributions are characterized by having diverging Laplace–Stieltjes transforms for all negative values of the argument θ, i.e. Z ∞

e−θs dH(s) = ∞.

0

Moreover, some of the most commonly used heavy–tailed distributions do not have Laplace–Stieltjes transforms with closed–form expressions; that is the case of the Lognormal, Fréchet, Gumbel and Weibull distributions. This implies that classical approaches employed for approximating tail probabilities of convolutions are not available or somewhat restricted. For instance, large deviations techniques, saddlepoint approximations, exponential changes of measure, or the applications of the Pollaczek–Khinchine formula in risk and queueing, require of the Laplace–Stieltjes transform in closed form. A variety of approaches do exist for dealing with stochastic models having heavy– tailed inputs. Among these we are interested in approximating heavy–tailed distributions within certain classes of tractable distributions. An obvious candidate is the class of phase–type distributions which is dense in the nonnegative distributions (cf. Asmussen, 2003), so one could in principle approximate any nonnegative heavy– tailed distribution with an arbitrary precision. The class of phase–type distributions possesses many attractive features: expressions for densities, cumulative distribution functions, moments and integral transforms have closed–form expressions in terms of matrix exponential functions; in addition, the class is closed under finite mixtures and convolutions (Assaf and Levikson, 1982; Maier and O’Cinneide, 1992; Neuts, 2

1975). This classical approach has been widely studied, and reliable methodologies for approximating any nonnegative theoretical distribution with a phase–type distribution are already available (cf. Asmussen et al., 1996). However, phase–type distributions are inherently light–tailed so these are limited to the Gumbel domain of attraction (Kang and Serfozo, 1999). Therefore, in spite of its denseness, phase–type distributions cannot correctly capture the characteristic behavior of a heavy–tailed distribution and these may deliver unreliable approximations for certain quantities of interest (Vatamidou et al., 2012, 2014). An alternative was recently analyzed in Bladt et al. (2014). The authors consider a subclass of phase–type scale mixtures as defined in (1.1), but restricted to having a discrete scaling distribution H. Such distributions correspond to distribution of absorptions times of Markov jump processes with a countable number of transient states. Clearly, such a class not only contains all phase–type distributions but shares many of its properties; for instance, it is a dense class in the nonnegative distributions. It also contains a very large subclass of heavy–tailed distributions. The idea of extending the class of phase–type distributions to include heavy–tailed distributions by allowing an infinite number of transient states is well known among the Matrix Analytic community and attributed to Neuts (1981). Nevertheless, the class is not yet fully documented and to the best of the authors’ knowledge the only published reference available is Ding Shi et al. (1996), who defined the class of infinite dimensional phase–type distributions and denoted SPH–distributions. Another reference of interest is Greiner et al. (1999), who considered infinite mixtures of exponential distributions to approximate power–tailed distributions (eg. Pareto distributions). Nevertheless, Greiner assumed a bounded support of the scaling distribution H, but as we will see later, such restriction implies that such a mixture distribution is light–tailed. A general method for approximating heavy–tailed distributions via phase–type scale mixtures is not yet available, neither is their tail behavior fully understood. In this paper we concentrate on the second open problem. Our purpose is to characterize the tail behavior of phase–type scale mixture distributions, and provide conditions that will allow us to classify their maximum domains of attraction. In certain cases of interest we can obtain the exact tail asymptotics under very general conditions. We expect our results to be useful for selecting the scaling distribution H of a phase– type scale mixture, which in turn will be used to approximate general heavy–tailed distributions. Our contributions are as follows. Firstly, we prove that if the scaling distribution H has unbounded support, then the corresponding phase–type scale mixture distribution is heavy–tailed. This is of particular interest because the class of phase–type distributions is inherently light–tailed ; in contrast, when multiplied with a random variable S with unbounded support, it becomes heavy–tailed regardless of whether the distribution of S is either light or heavy–tailed. In addition we provide sufficient conditions for a mixture of general light–tailed distributions (not necessarily phase–type) to be either light or heavy–tailed. We note that a similar phenomenon is observed in Asmussen et al. (2008), where the total time of a job that must restart when a failure occurs is modeled via random sums. They show that if the task time distribution has an unbounded support then the total time distribution is heavy– 3

P tailed. Notice that instead of considering the distribution of a random sum Si=1 Yi , we look at the distribution of the product S · Y . Therefore, it is not surprising to draw analogue conclusions about the tail behavior under similar assumptions on the underlying random variables Y, Y1 , Y2 , . . . and S. Secondly, we determine the maximum domain of attraction of the class of phase– type scale mixture distributions. For scaling distributions in the Fréchet domain of attraction H ∈ MDA(Φα ), Breiman’s lemma implies that if the α–moment of G is finite then the phase–type scale mixture F remains in the Fréchet domain of attraction with the same index. More precisely, if H ∈ R−α =⇒ F (x) ∼ MG (α)H(x),

x → ∞,

where MG (α) is the α–moment of G. A number of extensions and converses of Breiman’s lemma had been proved under general assumptions for the distributions G and H (for a review, see Jessen and Mikosch, 2006, and references therein). Here, we show that the converse of Breiman’s lemma holds if G is a phase–type distribution, i.e. H ∈ MDA(Φα ) ⇐⇒ F ∈ MDA(Φα ). (1.2) Our proof requires a Tauberian theorem, which relates the Laplace–Stieltjes transform of the reciprocal of the scaling random variable L1/S (θ) := E[e−θ/S ] with the tail behavior of the phase–type scale mixture distribution F . In addition we prove a partial analogue of (1.2). We show that if a certain higher order derivative of L1/S (θ) is a von Mises function, then F ∈ MDA(Λ). The rest of the paper is organized as follows. In Section 2 we set up notation and summarize without proofs some of the standard facts on heavy–tailed and phase– type distributions. Then we introduce the class of phase–type scale mixtures and examine some of its asymptotic properties. Our main results on the tail behavior of phase–type scale mixtures are presented and proved in Section 3. Section 4 is devoted to discrete scaling distributions. In Section 5 we present our conclusions and discuss further directions of research.

2

Preliminaries

In this Section we recall several fundamental concepts needed for the development of our results. First we discuss heavy–tailed distributions, maximum domains of attraction and subexponentiality. Then we revise the asymptotic properties of phase– type distributions. Finally, we introduce the class of phase–type scale mixtures which is the main object of study in this paper.

2.1

Heavy–Tailed Distributions

Definition 2.1. We say that a nonnegative distribution H is heavy–tailed iff Z e−θs dH(s) = ∞, ∀θ < 0. Otherwise, we say that H is a light–tailed distribution. 4

The definition above is equivalent to having lim sup H (s)eθs = ∞,

∀θ > 0,

s→∞

where H (s) = 1−H(s) is the tail probability of the distribution H (Foss et al., 2011). We are also interested in the Fréchet and Gumbel domains of attraction (Fisher and Tippett, 1928; Gnedenko, 1943). A convenient way of characterizing those is via regularly varying and von Mises functions respectively (de Haan, 1970). Definition 2.2 (Regularly varying). A function h is regularly varying with index β ∈ R (h ∈ Rβ ) iff h(st) = tβ , t > 0. (2.1) lim s→∞ h(s) Otherwise, if the limit above is 0 for all t > 1, then we say that h is of rapid variation and we denote it h ∈ R−∞ (cf. Bingham et al., 1987). We say H is a regularly varying distribution with index α > 0 if H ∈ R−α . Regular variation characterizes the Fréchet domain of attraction via the following relation H ∈ R−α ⇐⇒ H ∈ MDA(Φα ). That is, H belongs to the Fréchet domain of attraction iff H is a regularly varying distribution (de Haan, 1970). The characterization of the Gumbel domain of attraction is more involved. A main result in extreme value theory indicates that a distribution H belongs to the Gumbel domain of attraction iff H is a von Mises function. The following result provides a characterization for distributions which are von Mises functions: Theorem 2.3 (de Haan (1970)). Let H be a twice differentiable nonnegative distribution with unbounded support. Then H is a von Mises function iff there exists s0 such that H 00 (s) < 0 for all s > s0 , and H (s)H 00 (s) = −1. s→∞ (H 0 (s))2 lim

Moreover, von Mises functions are functions of rapid variation (cf. Bingham et al., 1987; Embrechts et al., 1997). We are also interested in determining whether a distribution H in a maximum domain of attraction is subexponential. Definition 2.4 (Subexponential distribution). We say that H belongs to the class of subexponential distributions, denoted H ∈ S, iff lim sup s→∞

H ∗n (s) = n, H (s)

where H ∗n is the tail probability of the n-fold convolution of H.

5

The Fréchet case is straightforward because regularly varying distributions are subexponential, i.e. MDA(Φα ) ⊂ S for all α > 0. For the Gumbel domain of attraction, Goldie and Resnick (1988) provide a sufficient condition for a distribution H ∈ MDA(Λ) to be subexponential: Theorem 2.5 (Goldie and Resnick (1988)). Let H ∈ MDA(Λ) be an absolutely continuous function with density h, then H ∈ S if lim inf s→∞

H (ts) h(s) > 1, h(ts) H (s)

for all t > 1.

2.2

Phase–Type Distributions

A phase–type distribution corresponds to the distribution of the absorption time of a Markov jump process {Xt }t≥0 with a finite state space E = {0, 1, 2, · · · , p}. The states {1, 2, · · · , p} are transient while the state 0 is an absorbing one. We will consider proper distributions only so the Markov jump process cannot be started in the absorbing state. Hence, phase–type distributions are characterized by a pdimensional row vector β = (β1 , · · · , βp ) (corresponding to the probabilities of starting the Markov jump process in each of the transient states), and an intensity matrix   0 0 Q= . λ Λ Here Λ is a sub–intensity matrix of dimension p, λ a p-dimensional column vector and 0 the p-dimensional zero row vector (Asmussen, 2003; Latouche and Ramaswami, 1999, cf.). Since the rows in an intensity matrix must sum to 0, we have λ = −Λe, where e is a p-dimensional column vector of ones. Then we let Y be the time until absorption Y = inf{t > 0 : Xt = 0}. The distribution of Y is called a phase–type distribution with parameters (β, Λ) and we write Y ∼ PH(β, Λ). Here we are interested in the distribution of the scaled phase–type random variable sY where s > 0, so it follows that sY ∼ PH(β, Λ/s). Hence, the distribution function of sY is given by Gs (x) := G(x/s) = 1 − βeΛx/s e,

∀x > 0,

while its density is

Here, the function eΛx/s

λ (2.2) gs (x) = g(x/s) = βeΛx/s . s denotes a matrix exponential function which is defined as Λx/s

e

=

∞ X Λ k xk k=0

6

sk k!

.

We also recall that the n-th moment of sY ∼ PH(β, Λ/s) is given by Z ∞ (−1)n n!βΛ−n e MG (n) := tn dGs (t) = , ∀n ∈ N sn 0 (cf. Latouche and Ramaswami, 1999). Proposition 2.6. Let Gs ∼ PH(β, Λ/s). The tail probability of Gs can be written as   j −1  m ηX X x k 0,

(2.5)

i=1

where e is a finite dimension column vector while β is a finite dimension row vector representing an initial distribution.

3

Main results

Next we present our main results. We are interested in determining the tail properties of a phase–type scale mixture distribution F with scaling distribution H and phase–type distribution G. Recall that such a mixture also corresponds to the distribution of a product of two random variables S · Y where Y has a phase–type distribution G and S has a nonnegative but otherwise arbitrary distribution H. For our developments below it will often be convenient to use the second interpretation. For the rest of the paper, we assume that the eigenvalues of the sub-intensity matrix Λ are real unless stated otherwise. Therefore the tail probability of the corresponding phase–type scale mixture distribution will have the form F (x) =

j −1 Z m ηX X

j=1 k=0

∞

cjk

0

9

 x k s

e−λj x/s dH(s).

3.1

Tail behavior of phase–type scale mixtures

In the following theorem we provide a simple condition on the support of the scaling distribution H which implies that the phase–type scale mixture is heavy–tailed. Theorem 3.1. A phase–type scale mixture distribution is heavy–tailed iff its scaling distribution H has unbounded support. Proof. Assume that H has unbounded support. Then for any θ > 0 we can choose  ∈ (0, θ) and take s∗ in the support of H and large enough such that s∗ ≥ λj (θ−)−1 , j = 1, . . . , m. Notice that θ − λj /s ≥  for all s ≥ s∗ , so it follows that  Z s∗  Z ∞ θx F (x)e = G (x/s)dH(s) + G (x/s)dH(s) eθx s∗

0

≥

j −1 m ηX X

Z

s∗

j=1 k=0

≥

j −1 m ηX X

∞

cjk Z

∞

cjk s∗

j=1 k=0

 x k s  x k s

e(θ−λj /s)x dH(s)

ex dH(s).

Lebesgue’s monotone convergence implies that lim F (x)eθx = ∞,

x→∞

∀θ > 0.

Hence, F is a heavy–tailed distribution. To prove the converse we assume that H has bounded support with right endpoint sH = inf{s : H (s) > 0}. Then by stochastic dominance we have that F (x) = P(S · Y > x) ≤ P(sH · Y > x) = G (x/sH ). But G last is light–tailed, so we arrive at a contradiction. Hence, if a phase–type scale mixture F is heavy–tailed then its scaling distribution must have unbounded support. In the following example we assume both G and H have (light–tailed) exponential distributions so the hypotheses of theorem 3.1 are satisfied, hence the phase–type scale mixture distribution is heavy–tailed. In this simple case we can calculate the exact asymptotics. Example 3.2. (Exponential Mixtures) Let G ∼ exp(λ) and H ∼ exp(β). Then the tail probability of the phase–type scale mixture distribution F has the form Z ∞ Z ∞ p p F (x) = G (x/s)dH(s) = β e−λx/s−βs ds = 2 βλx BesselK(1, 2 βλx), 0

0

where BesselK is the modified function of the second kind. p Bessel −x Since BesselK(·, x) ∼ π/(2x)e (see Appendix A), it follows that √ 2π −2√λβx+θx lim eθx F (x) = lim e = ∞, ∀θ > 0. x→∞ (λβx)1/4 x→∞ 10

The conclusion of theorem 3.1 can be extended to a wider class beyond phase– type scale mixture distributions. The following theorem provides sufficient conditions for the distribution of the mixture of two unbounded light–tailed distributions (not necessarily phase–type) to be either light or heavy–tailed. Theorem 3.3. Consider H1 and H2 two nonnegative distributions with unbounded support and let H be the mixture distribution of H1 and H2 . 1. If there exist θ > 0 and ξ(x) a nonnegative function such that   lim sup eθx H 1 (x/ξ(x)) + H 2 (ξ(x)) = 0,

(3.1)

x→∞

then H is a light–tailed distribution. 2. If there exists ξ(x) a nonnegative function such that for all θ > 0 it holds that lim sup eθx H 1 (x/ξ(x)) · H 2 (ξ(x)) = ∞,

(3.2)

x→∞

then H is a heavy–tailed distribution. Proof. For the first part consider Z ∞ θx θx lim sup H (x)e = lim sup e H 1 (x/s)dH2 (s) x→∞ x→∞ 0  Z ξ(x)  Z ∞ θx θx = lim sup e H 1 (x/s)dH2 (s) + e H 1 (x/s)dH2 (s) x→∞ 0 ξ(x)   ≤ lim sup eθx H 1 (x/ξ(x)) + eθx H 2 (ξ(x)) = 0. x→∞

The last equality holds by the hypothesis (3.1). Hence H is light–tailed. For the second part consider  Z ξ(x)  Z ∞ θx θx θx H 1 (x/s)dH2 (s) + e H 1 (x/s)dH2 (s) lim sup H (x)e = lim sup e x→∞ x→∞ 0 ξ(x)   ≥ lim sup eθx H 1 (x/ξ(x))H 2 (ξ(x)) = ∞. x→∞

The last equality holds by hypothesis (3.2). Hence H is heavy–tailed. In some practical cases the conditions in theorem 3.3 can be used to completely classify the tail behavior of a product of two random variables with distributions in a specific parametric family. For instance, the following example demonstrates rather strikingly that the result in theorem 3.3 is enough to fully classify the tail behavior of a product of two Weibull random variables. Example 3.4. (Weibull Mixtures) Let H1 ∼ Weibull(λ, p) and H2 ∼ Weibull(β, q). Then the mixture distribution H of H1 and H2 is light–tailed iff 1 1 + < 1. p q Otherwise it is heavy–tailed. 11

(3.3)

Proof. Recall that p

H 1 (s) = e−(s/λ) , H 2 (s) = e

−(s/β)q

s ≥ 0, s ≥ 0.

,

Let us first assume that (3.3) holds. Then we can choose γ ∈ (q −1 , 1 − p−1 ) so we obtain the inequalities (1 − γ)p > 1 and γq > 1. Taking ξ(x) = xγ and θ > 0 we get     1−γ p γ q lim eθx H 1 (x/ξ(x)) + H 2 (ξ(x)) = lim eθx−(x /λ) + eθx−(x /β) = 0. x→∞

x→∞

Case 1 of theorem 3.3 implies the distribution F is light–tailed. Next, suppose that condition (3.3) does not hold so we can choose γ ∈ (1 − p−1 , q −1 ) to obtain the reversed inequalites inequalities (1 − γ)p < 1 and γq < 1. Taking ξ(x) = xγ and θ > 0 we obtain   1−γ p γ q lim eθx H 1 (x/ξ(x))H 2 (ξ(x)) = lim eθx−(x /λ) −(x /β) = ∞, ∀θ > 0. x→∞

x→∞

Case 2 of theorem 3.3 holds, so the distribution H is heavy–tailed. Remark 3.5. Note that if we let p = q we can compute explicitly the tail behavior of the product of two Weibull random variables. Z ∞ Z ∞ p p H (x) = H 1 (x/s)dH2 (s) = e−(x/λs) p β −p sp−1 e−(s/β) ds 0

0 p 2

p

= 2(x/λβ) BesselK(1, 2(x/λβ) 2 ) ∼

√

p

p

π(x/λβ) 4 e−2(x/λβ) 2 .

Next observe that lim eθx H (x) = lim

x→∞

x→∞

√

p

p

π(x/λβ) 4 eθx−2(x/λβ) 2 =

  ∞, p < 2, 0 < θ,  

0,

p

2 ≤ p, 0 < θ < 2(λβ)− 2 .

Hence, H is heavy–tailed iff p < 2; this is equivalent to condition (3.3).

3.2

Fréchet Domain of Attraction

Next we investigate the maximum domain of attraction of phase–type scale mixtures. The main result for the Fréchet domain of attraction is as follows: Theorem 3.6 (Fréchet Domain of Attraction). Let F be a phase–type scale mixture with scaling distribution H and phase–type distribution G. Then H ∈ MDA(Φα ) ⇐⇒ F ∈ MDA(Φα ).

(3.4)

Moreover, in such a case the asymptotic behavior of the distributions F and H are related via the identity F (x) = MG (α)H (x)(1 + o(1)),

x → ∞,

where MG (α) is the α–moment of the phase–type distribution G. 12

(3.5)

Breiman’s lemma (Breiman, 1965) states that if H ∈ MDA(Φα ) and MG (α+) < ∞, then F ∈ MDA(Φα ) and the asymptotic relation (3.5) holds. We provide a simplified proof of Breiman’s lemma when G is a phase–type distribution. We emphasize that the converse of Breiman’s lemma is not true in general so the proof of theorem 3.6 is non–trivial. Jessen and Mikosch (2006) provide a complete list of references with sufficient conditions for the converse to hold true and we note that the case considered here is not fully covered in there. Our contribution is to prove that the converse holds true if G is a phase–type distribution having a sub– intensity matrix with real eigenvalues. We require a particular Tauberian theorem on regular variation which establishes a relation between a tail probability and the Laplace–Stieltjes transform of the reciprocal. This approach will shed some new light into the asymptotic behavior of phase–type scale mixtures. The proof of theorem 3.6 requires several lemmas which are of interest on their own and given next. Recall that a distribution F belongs to the Fréchet domain of attraction with index α iff and only if F ∈ R−α . Lemma 3.7 (Breiman’s lemma). Assume F , G and H satisfy the assumptions of theorem 3.6. If H ∈ R−α then x → ∞.

F (x) = MG (α)H (x)(1 + o(1)),

Proof. Recall that H(x) = L(x)/xα for some slowly varying function L(x) at infinity. Hence the tail probability F (x) can be written as Z ∞ Z ∞ L(x/t) F (x) = H (x/t)dG(t) = dG(t). (x/t)α 0 0 Then F (x) lim = lim x→∞ H (x) x→∞

R∞ 0

L(x/t) dG(t) (x/t)α L(x)/xα

Z = lim

x→∞

0

∞

L(x/t) α t dG(t). L(x)

The result of the lemma follows by dominated convergence and the fact that all moments of G are finite. The following lemma states that a phase–type scale mixture is in the Fréchet domain of attraction iff the Laplace–Stieltjes transform of S −1 is regularly varying, with S ∼ H. Lemma 3.8. Assume F , G and H satisfy the assumptions of theorem 3.6. Then F ∈ R−α

⇐⇒

L1/S ∈ R−α ,

where S ∼ H and L1/S is the Laplace–Stieltjes transform of S −1 . Proof. Observe that the distribution function F can be written as Z ∞  k j −1 m ηX X x F (x) = e−λj x/s dH(s) cjk s 0 j=1 k=0 =

j −1 m ηX X

j=1 k=0

cjk

(−1)k xk λkj 13

Z 0

∞

dk −λj x/s e dH(s). dxk

The conditions in theorem 16.8 in Billingsley (1995) are satisfied so we can exchange the integral and derivative operations to obtain F (x) =

j −1 m ηX X

j=1 k=0

cjk

(−1)k xk dk L1/S (λj x). dxk λkj

dk L1/S (x) ∈ R−α−k . Next observe that if F (x) ∈ R−α , then ∃k ∈ N such that dxk Therefore, Karamata’s theorem implies that for all k it holds that xk

dk L1/S (x) ∈ R−α dxk

⇐⇒

L1/S (x) ∈ R−α .

This completes the proof. The next is a Tauberian result which characterizes regular variation via the Laplace–Stieltjes transform. The proof, which is omitted, follows from Breiman’s lemma in one direction and Karamata’s Tauberian theorem for the converse (cf. Mao and Hua, 2014, Theorem 4.1) Lemma 3.9. Let S ∼ H and let α > 1. Then L1/S ∈ R−α

⇐⇒

H ∈ R−α .

The proof of theorem 3.6 is given next. Proof of theorem 3.6. ⇒ Assume that H ∈ R−α , Breiman’s lemma 3.7 implies that F ∈ R−α . ⇐ Assume that F ∈ R−α , lemma 3.12 implies that L1/S ∈ R−α . Using lemma 3.9 we obtain S ∼ H ∈ R−α . The following corollary provides expressions for the norming constants. It shows that the norming constants for a phase–type scale mixture distribution F can be chosen as the norming constants of H divided by the α-moment of the phase–type distribution G. Corollary 3.10. Assume that H is regularly varying with index α, then the norming constants can be chosen as  ← 1 1 dn = 0, cn = (n). MG (α) H Proof. From theorem 3.6, F (x) ∼ MG (α)H (x),

x → ∞.

Recall that a set of norming constants for a distribution F in the Fréchet domain of attraction is given by  ← 1 dn = 0, cn = (n) F 14

(Embrechts et al., 2014). Thus,  ←  ← 1 1 1 cn = (n) = (n). MG (α) H MG (α)H

In the following example we consider a Pareto scaling distribution with index α. Note that we employ a nonstandard parametrization which is convenient for calculating the exact asymptotics of the associated phase–type scale mixture distribution. Example 3.11 (Pareto scaling). Consider H (s) = s−α with s ≥ 1. Then the phase– type scale mixture F with scaling distribution H is heavy–tailed and belongs to the Fréchet domain of attraction. Proof. Observe that a change of variable w = λj x/s and Lebesgue’s monotone convergence imply that m ηj −1

XX F (x) lim −α = lim xα cjk x→∞ x→∞ x j=1 k=0 = lim

=

s

λj x

e−λj x/s

α sα+1

ds

wk+α e−w dw

0

j −1 m ηX X cjk αΓ(α + k)

j=1 k=0

 x k

1

Z j −1 m ηX X cjk α

x→∞ λk+α j=1 k=0 j

∞

Z

λα+k j

.

Hence F (x) ∈ MDA(Φα ). The norming constants can be chosen as  dn = 0,

cn =

1 G

← (n) =

j −1 m ηX X cjk αΓ(α + k)

j=1 k=0

!1/α

λα+k j

n−1/α .

A case of interest is G ∼ exp(λ). It is readily verified that m = 1, k = 0, c10 = 1 so F (x) =

3.3

αΓ(α) −α α! −α x = x = MG (α)x−α . λα λα

The Gumbel domain of attraction

Next we turn our attention to the Gumbel domain of attraction. theorem 3.12 below provides a sufficient condition for a phase–type scale mixture to belong to the Gumbel domain of attraction. Such a result says that if a certain higher order derivative of the Laplace–Stieltjes transform of S −1 is a von Mises function, then the phase–type scale mixture is in the Gumbel domain of attraction.

15

Theorem 3.12. Assume F , G and H satisfy the assumptions of theorem 3.6. Let (η−1) V (x) = (−1)η−1 L1/S (x) where η is the largest dimension among the Jordan blocks associated to the largest eigenvalue of the sub–intensity matrix. If V (·) is a von Mises function, then F ∈ MDA(Λ). Moreover, if lim inf x→∞

V (tx)V 0 (x) > 1, V 0 (tx)V (x)

∀t > 1,

then F is subexponential. Proof. Recall that F ∈ MDA(Λ) iff F (x)F 00 (x) = −1 x→∞ (F 0 (x))2 lim

(3.6)

(de Haan, 1970). In addition, let a(x) = F (x)/f (x) which is called an auxiliary function of F , then if a(tx) lim inf > 1, ∀t > 1, (3.7) x→∞ a(x) then F is subexponential (Goldie and Resnick, 1988). Hence, we shall prove that the hypotheses of the theorem 3.12 imply (3.6) and (3.7). Recall that we can write F (x) =

j −1 m ηX X

j=1 k=0

cjk

(−1)k xk dk L1/S (λj x). dxk λkj

(η−1)

Since V (x) = (−1)η−1 L1/S (x) is a von Mises function, then V (x) is of rapid variation (Bingham et al., 1987; Embrechts et al., 1997). This implies that F (x) ∼ c

xη−1 V (λx), λη−1

(3.8)

where c is some constant, −λ is the largest eigenvalue of the sub–intensity matrix and η is the largest dimension among the Jordan blocks associated to −λ. In addition 0

F (x) = −

j −1 m ηX X

 cjk

 (−1)k kxk−1 dk (−1)k xk dk+1 L1/S (λj x) + L1/S (λj x) dxk dxk+1 λkj λk−1 j

j=1 k=0 η−1 η−1

∼ −c

(−1) x λη−2

dη xη−1 0 L (λx) = −c V (λx), 1/S dxη λη−2

and 00

F (x) = −

j −1 m ηX X

j=1

∼ −c

(−1)k k(k − 1)xk−2 dk (−1)k kxk−1 dk+1 L (λ x) + L (λj x) j dxk 1/S dxk+1 1/S λkj λjk−1 k=0  (−1)k kxk−1 dk+1 (−1)k xk dk+2 + L (λj x) + L (λj x) dxk+1 1/S dxk+2 1/S λk−1 λk−2 j j 

cjk

(−1)η−1 xη−1 dη+1 xη−1 00 L (λx) = −c V (λx). 1/S λη−3 dxη+1 λη−3

16

Hence it follows that V (x)(−V 00 (x)) F (x)F 00 (x) = lim
2 = −1. x→∞ x→∞ (F 0 (x))2 − V 0 (x) lim

(η−1)

The last holds true because by hypothesis V (x) = (−1)η−1 L1/S (x) is a von Mises function. Hence F ∈ MDA(Λ) and the first part result follows. For the second part, we observe that the auxiliary function a(x) = F (x)/f (x) obeys the following asymptotic equivalence a(x) =

F (x) V (x) ∼ . f (x) −V 0 (x)

The distribution F is subexponential iff lim inf x→∞

V (tx)V 0 (x) a(tx) = lim inf 0 > 1, x→∞ V (tx)V (x) a(x)

∀t > 1,

hence subexponentiality of F follows.

Example 3.13 (Exponential scaling). Let H ∼ exp(β). Then F is a subexponential distribution in the Gumbel domain of attraction. Proof. Observe that 1/S has an inverse Gamma distribution with a Laplace–Stieltjes transform given in terms of a modified Bessel function of the second kind: Z ∞ p p e−λx/s βe−βs ds = 2 λβxBesselK(1, 2 λβx). L1/S (x) = 0

Asymptotically it holds true that √ √ 1 (n) L1/S (x) ∼ (−1)n π(λβx) 4 e− λβx .

Hence, it follows that

V (x)(−V 00 (x)) = −1. x→∞ (−V 0 (x))2 lim

Therefore V (x) is a von Mises function and F ∈ MDA(Λ). Moreover, if t > 1 then a(tx) V (tx)V 0 (x) √ = lim 0 = t > 1. x→∞ a(x) x→∞ V (tx)V (x) lim

Thus F is subexponential distribution. Remark 3.14. Notice that it is possible to generalize the result of the previous example for Gamma scaling distributions because an expression for the Laplace– Stieltjes transform of an inverse gamma distribution is known and given in terms of a modified Bessel function of the second kind. However, it involves a number of tedious calculations and therefore omitted. Note as well that in such a case it is possible to test directly if F is a von Mises function, but the calculations become cumbersome. 17

Example 3.15 (Lognormal scaling). Assume H ∼ LN(0, σ), then F is a subexponential distribution in the Gumbel domain of attraction. Proof. First observe that the symmetry of the normal distribution implies that the Laplace–Stieltjes transform of 1/S is the same as that of S, i.e. L1/S (x) = LS (x). An asymptotic approximation of the k-th derivative of the Laplace–Stieltjes transform of the lognormal distribution is given in Asmussen et al. (2015): 1 (k) LS (x) = (−1)k LS (x) exp{−kω0 (x) + σ0 (x)2 k 2 }(1 + o(1)), 2 where

σ2 , 1 + ωk (x) and W(·) the Lambert W function. Hence we verify that 2

ωk (x) = W(xσ 2 ekσ ),

σk (x)2 =

1

2

1

2

2 (η+1)2

V (x)V 00 (x) e−(η−1)ω0 (x)+ 2 σ0 (x) (η−1) · e−(η+1)ω0 (x)+ 2 σ0 (x) lim = lim x→∞ x→∞ V 0 (x)2 e−2ηω0 (x)+σ0 (x)2 η2   σ2 2 = lim exp{σ0 (x) } = lim exp . x→∞ x→∞ 1 + ω0 (x)

As ωk (x) is asymptotically of order log(x) as x → ∞, then σ 2 (1 + ω0 (x))−1 → 0 as x → ∞. Then the last limit is equal to 1 so we have shown that F (x) ∈ MDA(Λ). Furthermore, (η)

(η−1)

(−1)η−1 L1/S (tx) · (−1)η L1/S (x) a(tx) = lim lim x→∞ (−1)η L(η) (xt) · (−1)η−1 L(η−1) (x) x→∞ a(x) 1/S 1/S 1

= lim

2 (η−1)2

e−(η−1)ω0 (xt)+ 2 σ0 (xt) 1

1

1

2 η2

2

2

· e−(η−1)ω0 (x)+ 2 σ0 (x) (η−1)  1 1 2 2 = lim exp −ω0 (x) + ω0 (xt) + σ0 (xt) (2η − 1) + σ0 (x) (1 − 2η) x→∞ 2 2  −1 = lim exp −ω0 (x) + ω0 (x) + ω0 (t) + O(ω0 (x) ) = t > 1. x→∞

0 (xt)+ 2 σ0 (xt) e−ηω

2 η2

· e−ηω0 (x)+ 2 σ0 (x)

x→∞

Thus F is a subexponential distribution. Remark 3.16. We note that in the non Fréchet case, the distribution of F will typically have a very different tail behavior than that of the scaling distribution H (this is in contrast to the Fréchet case where the tail probability of F is asymptotically proportional to the tail probability of H). As an example we consider the lognormal case H ∼ LN(0, σ 2 ). Using relation (3.8) and Mill’s ratio we obtain that the asymptotic ratio of F and H is given by (η−1)

F (x) cxη−1 LS (x) lim = x x→∞ H (x) h(x) log(x) = lim c1 x x→∞

η−1

 log x exp

−wk2 (x) − 2wk (x) + log2 (x) 2σ 2

18

 = ∞.

The last limit follows since W(x) ≥ log x − 2−1 log log x. Since the phase–type scale mixture distribution has a much heavier tail than the lognormal, it seems necessary to search for other alternative scaling distributions to obtain better approximations of the lognormal via phase–type scale mixtures.

4

Discrete Scaling Distributions

In this Section we will consider discrete scaling random variables. Such cases are of major interest as their associated phase–type scale mixture distributions correspond to distributions of absorption times of Markov jump processes with an infinite number of states. Discrete distributions often involve infinite convergent series with unknown limits. Hence, the situation is typically more involved than in the continuous case. Here we suggest a simple method to approximate such infinite series via their analogue integrals. Let us assume that the scaling random variable is supported over some discrete set {s1 , s2 , . . . } for which there exists a nonnegative continuous function s : R+ → R+ with s(i) = si . Let pi = P(S = si ) and further assume that pi = p(i) for some nonnegative continuous function p(·). Then the phase–type scale mixture associated with the distribution of the random variable S is given by F (x) =

∞ X

p(i)G(x/s(i)).

i=1

Define g(y; x) = p(y)G(x/s(y)). If for all x > 0, it holds that g(y; x) is Riemann integrable and has a single local maxima at yb, then we can bound the infinite sum with: Z ∞ Z ∞ ∞ X g(y; x)dy + g(b y ; x). g(y; x)dy − g(b y ; x) ≤ g(i; x) ≤ 0

0

i=1

For an heuristic argument justifying the previous inequalities see figure 1. The idea of the method is to compare the asymptotic order of the integral and the maxima as x → ∞. If the value of the maxima g(b y ; x) is asymptotically negligible with respect to the value of the integral then it follows that the distribution function is asymptotically equivalent to the value of the integral. Example 4.1 (Zipf scaling). Let α ≥ 2 and assume H ∼ Zipf(α), which is called the Zipf or Zeta distribution. The Zipf distribution is supported over the natural numbers and defined by p(i) = i−α /ξ(α) where ξ(·) is the Riemann zeta function. Then F is a heavy–tailed distribution in the Fréchet domain of attraction. Proof. The tail probability of F is given by F (x) =

j −1 ∞ X m ηX X

cjk

 x k

i=1 j=1 k=0

=

i

j −1 ∞ m ηX X X cjk xk

j=1 k=0 i=1

ξ(α)

19

e−λj x/i

i−α ξ(α)

i−(α+k) e−λj x/i .

Figure 1: Bound infinite sum with integral

Consider the functions gjk (y; x) = xk y −(α+k) e−λj x/y and note that each of these functions attains its single local maximum at yˆ = λj x(α + k)−1 > 0, for all x > 0. Therefore Z ∞ Z ∞ ∞ X k −(α+k) −λj x/i gjk (y; x)dy − gjk (ˆ y ; x) ≤ x i e ≤ gjk (y; x)dy + gjk (ˆ y ; x). 0

0

i=1

Observe that 

k −(α+k)

gjk (ˆ y ; x) = x e and

λj α+k

−(α+k)

x−(α+k) = cx−α ,

∞

Γ(α + k − 1) −α+1 x , λα+k−1 0 so gjk (ˆ y ; x) is of negligible order with respect to Ijk (x). Then it follows that Ijk (x) := x

k

Z

y −(α+k) e−λj x/y dy =

j −1 m ηX X cjk F (x) ∼ Ijk (x), ξ(α) j=1 k=0

=

x → ∞,

j −1 m ηX X cjk Γ(α + k − 1)

j=1 k=0

Thus F (x) ∈MDA(Φα−1 ). Let C =

ξ(α)λα+k−1

x−α+1 ,

x → ∞.

j −1 m ηP P cjk Γ(α + k − 1) , then the norming conξ(α)λα+k−1 j=1 k=0

stants can be chosen as  dn = 0,

cn =

1 F

←

20

1   α−1 C . (n) = n

Example 4.2 (Geometric scaling). Let H ∼ Geo(p), then F is a subexponential distribution in the Gumbel domain of attraction. Proof. We let p(i) = pq i where q = 1 − p. Since the geometric distribution has unbounded support, then the associated phase–type scale mixture is heavy–tailed. We next verify that it belongs to the Gumbel domain of attraction. F (x) =

j −1 ∞ X m ηX X

cjk

i=1 j=1 k=0

=

j −1 m ηX X

pcjk xk

j=1 k=0

 x k i

e−λj x/i pq i

∞ X e−λj x/i q i i=1

ik

.

Let gjk (y; x) = exp{−λj x/y + y log q − k log y} and yˆ be the value maximizing it. It is possible to show that √ x → ∞. gjk (ˆ y ; x) = x−k/2 e−2 λj | log q|x (c + o(1)), For the approximation of the integral, we obtain that   k−1 Z ∞ q | log q| 2 gjk (y; x)dy = 2 BesselK(k − 1, 2 λj x| log q|) λj x 0 √ x → ∞. = x−k/2+1/4 e−2 λj | log q|x (c + o(1)), So, the value of g(ˆ y ; x) is asymptotically negligible with respect to the value of the integral. Hence, Z ∞ −λj x/y y ∞ X e q e−λj x/i q i ∼ dy. k k i y 0 i=1 Then it follows that F (x) ∼ 2pcx

η−1



| log q| λx

 η−2 2

p BesselK(η − 2, 2 λx| log q|),

for some constant c. Similar argument yields   η−1 2 p d | log q| F (x) ∼ 2λpcxη−1 BesselK(η − 1, 2 λx| log q|), dx λx   η2 p d2 | log q| 2 η−1 λx| log q|). F (x) ∼ −2λ pcx BesselK(η, 2 dx2 λx All modified Bessel functions involved above are asymptotically equivalent, so it follows that F (x)F 00 (x) lim = −1, x→∞ (F 0 (x))2 so the distribution F belongs to the Gumbel domain of attraction. Moreover, it follows that for any t > 1, a(tx) F (tx)F 0 (x) √ = lim 0 = t > 1. x→∞ a(x) x→∞ F (tx)F (x) lim

Thus F is subexponential distribution. 21

5

Conclusion

We considered the class of phase–type scale mixtures. Such distributions arise from the product of two random variables S · Y , where S ∼ H is a nonnegative random variable and Y ∼ G is a phase–type distribution. Such a class is mathematically tractable and can be used to provide better approximations of heavy–tailed distributions. In this paper we focused on the analysis of the asymptotic behavior of their tail probabilities. We restricted our attention to phase–type scale mixtures having sub–intensity matrices with real eigenvalues, but we conjecture that our results remain true in the general case. Nevertheless, we point out that our results are highly relevant because the class of phase–type scale mixture distributions with real eigenvalues is dense in the nonnegative distributions. We proved that if the scaling distribution H has unbounded support, then the associated phase–type scale mixture distribution is heavy–tailed. In addition, we provided conditions for classifying the maximum domain of attraction of phase–type scale mixture distributions and explored subexponentiality. We proved a converse of Breiman’s lemma to show that the Fréchet domain of attraction is closed under scaling of phase–type random variables. In contrast, the Gumbel domain of attraction is more involved. We provided sufficient conditions on the Laplace–Stieltjes transform of the reciprocal of the scaling random variable for the corresponding phase–type scale mixture to be in the Gumbel domain of attraction. We noted in our examples that the tail probability of a phase–type scale mixture is much heavier than the tail of its corresponding scaling distribution. We also investigated subexponentiality. We provided sufficient conditions for a phase–type scale mixture to be subexponential. In particular, we were able to verify subexponentiality in all our examples.

Acknowledgements LRN is supported by Australian Research Council (ARC) grant DE130100819. WX is supported by IPRS/APA scholarship at The University of Queensland.

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A

Modified Bessel function of the second kind

A function playing a central role in our examples is the modified Bessel function (cf. Abramowitz and Stegun, 1964; Bowman, 1958): Definition A.1 (Modified Bessel functions). The second order differential equation z2

d2 ω dω + z − (z 2 + ν 2 )ω = 0, dz 2 dz

ν∈C

(A.1)

is called the modified Bessel differential equation. The solutions are called the modified Bessel functions of the first kind and the second kind respectively. 24

The modified Bessel function of the second kind has the following properties: 1. The Bessel function satisfies Z ∞ p p βe−xλ/s−βs ds = 2 λβx BesselK(1, 2 λβx). 0

2. The asymptotic behaviour of BesselK(ν, z) and its higher order derivatives is given by r π −z dn n e , z → ∞, BesselK(ν, z) ∼ (−1) dz n 2z for all n = 0, 1, . . . (Ragab, 1965).

25