Reflecting Brownian Motion in Two Dimensions: Exact Asymptotics for the Stationary Distribution

We consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 × 2 positive definite covariance matrix, and a 2 × 2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal stationary distribution along each direction in the quadrant. For a given direction, the marginal tail distribution has the exact asymptotic of the form bxκ exp(−αx) as x goes to infinity, where α and b are positive constants and κ takes one of the values −3/2, −1/2, 0, or 1; both the decay rate α and the power κ can be computed explicitly from the given direction and the SRBM data. A key tool in our proof is a relationship governing the moment generating function of the two-dimensional stationary distribution and two moment generating functions of the associated one-dimensional boundary measures. This relationship allows us to characterize the convergence domain of th...

We consider a two-dimensional semimartingale reflecting Brownian motion (SRBM) in the nonnegative quadrant. The data of the SRBM consists of a two-dimensional drift vector, a 2 × 2 positive definite covariance matrix, and a 2 × 2 reflection matrix. Assuming the SRBM is positive recurrent, we are interested in tail asymptotic of its marginal stationary distribution along each direction in the quadrant. For a given direction, the marginal tail distribution has the exact asymptotic of the form bx κ exp(−αx) as x goes to infinity, where α and b are positive constants and κ takes one of the values −3/2, −1/2, 0, or 1; both the decay rate α and the power κ can be computed explicitly from the given direction and the SRBM data.
A key tool in our proof is a relationship governing the moment generating function of the two-dimensional stationary distribution and two moment generating functions of the associated one-dimensional boundary measures. This relationship allows us to characterize the convergence domain of the two-dimensional moment generating function. For a given direction c, the line in this direction intersects the boundary of the convergence domain at one point, and that point uniquely determines the decay rate α. The one-dimensional moment generating function of the marginal distribution along direction c has a singularity at α. Using analytic extension in complex analysis, we characterize the precise nature of the singularity there. Using that characterization and complex inversion techniques, we obtain the exact asymptotic of the marginal tail distribution.
1. Introduction. This paper is concerned with the asymptotic tail behavior of the stationary distributions of two-dimensional semimartingale reflecting Brownian motions (SRBMs). As background for this study, we briefly discuss general multidimensional SRBMs. They are diffusion processes that arise as approximations for queueing networks of various kinds, cf. [12] and [30,31]. The state space for a d-dimensional SRBM Z = {Z(t), t ≥ 0} is R d + (the non-negative orthant). The data of the process are a drift vec-tor µ, a non-singular covariance matrix Σ, and a d × d "reflection matrix" R that specifies boundary behavior. In the interior of the orthant, Z behaves as an ordinary Brownian motion with parameters µ and Σ, and Z is pushed in direction R j whenever the boundary surface {z ∈ R d + : z j = 0} is hit, where R j is the jth column of R, for j = 1, . . ., d. To make this description more precise, one represents Z in the form Z(t) = X(t) + RY (t), t ≥ 0, (1.1) where X is an unconstrained Brownian motion with drift vector µ, covariance matrix Σ, and Z(0) = X(0) ∈ R d + , and Y is a d-dimensional process with components Y 1 , . . . , Y d such that Y is continuous and non-decreasing with Y (0) = 0, (1.2) Y j only increases at times t for which Z j (t) = 0, j = 1, . . ., d, and (1.3) The complete definition of the diffusion process Z will be given in Section A.1.
A d × d matrix R is said to be an S-matrix if there exists a d-vector w ≥ 0 such that Rw > 0 (or equivalently, if there exists w > 0 such that Rw > 0), and R is said to be completely-S if each of its principal sub-matrices is an S-matrix. (For a vector v, we write v > 0 to mean that each component of v is positive, and we write v ≥ 0 to mean that each component of v is nonnegative.) Taylor and Williams [29] and Reiman and Williams [27] show that for a given data set (Σ, µ, R) with Σ being positive definite, there exists an SRBM for each initial distribution of Z(0) if and only if R is completely-S. Furthermore, when R is completely-S, the SRBM Z is unique in distribution for each given initial distribution. A necessary condition of the existence of the stationary distribution for Z is (1.5) R is non-singular and R −1 µ < 0.
If R is an M-matrix as defined in Chapter 6 of [2], then (1.5) is known to be necessary and sufficient for the existence and uniqueness of a stationary distribution of Z; Harrison and Williams [13] prove that result and explain how the M-matrix structure arises naturally in queueing network applications. A square matrix is said to be a P-matrix if all of its principal minors are positive (that is, each principal submatrix of R has a positive determinant).
Obviously, M-matrix is a special case of P-matrix. For a two-dimensional SRBM, Harrison and Hasenbein [11] show that condition (1.5) and R being a P matrix are necessary and sufficient for the existence of a stationary distribution.
In this paper we are concerned with two-dimensional SRBMs. Throughput this paper, we assume R is a P matrix and (R, µ) satisfy (1.5). Therefore, such an SRBM has a unique stationary distribution. We are interested in the asymptotic tail behavior of the stationary distribution. Let Z(∞) ≡ (Z 1 (∞), Z 2 (∞)) be a random vector that has the stationary distribution of the SRBM. Let c ∈ R 2 + be a directional vector, i.e., a nonnegative vector in R 2 such that c ≡ c, c = 1, where x, y is the inner product of vectors x and y. We are interested in the asymptotic tail behavior of P{ c, Z(∞) ≥ x} as x → ∞. Our major interest is to compute exact asymptotics in any given direction c from the primitive data (Σ, µ, R). In this paper we will prove that, for each c ∈ R 2 + , f c (x) can be taken to be for some constant b > 0. That is, The exponent α c > 0 is known as the decay rate. The decay rate α c and the constant κ c can be computed explicitly from the primitive data, and κ c must take one of the values −3/2, −1/2, 0, or 1. The complete results are stated in Section 2.
Although our major interest is in the exact asymptotics of the tail probability, for many cases we have actually obtained the exact asymptotic for the density of the random variable c, Z(∞) . In these cases, it will be proven that for each direction c, the density p c (x) exists, p c (x) is continuous in x on [0, ∞), and as x → ∞, (1.8) with the same b, κ c and α c as in (1.7). In these cases, we first establish (1.8) and then obtain (1.7) from (1.8) as shown in Lemma D. 5. In other cases, we are not able to establish (1.8) and will work with the tail probabilities directly.
To get the exact asymptotics, we use the moment generating function of the random variable c, Z(∞) . Let ψ c (λ) = E e λ c,Z(∞) .
(This will be proved as a consequence of our Theorem 2.1.) Equivalently, the decay rate is the first singular point ψ c (z) on the real axis when ψ c (z) is viewed as a complex function of z. Let (1.9) ϕ(θ 1 , θ 2 ) = E e θ,Z(∞) be the two-dimensional moment generating function of Z(∞) = (Z 1 (∞), Z 2 (∞)).
Since ψ c (λ) = ϕ(λc) for λ ∈ R, the singularity of ϕ(zc) is used to determine the decay rate for each direction c. It turns out the singularity of ϕ(zc) allows us to apply a complex inversion technique to get the exact asymptotics.
To find the singularities of moment generating functions, one tries to derive closed form expressions of these functions. This is the approach used in [20]. That paper studied a tandem queue whose input is driven by a Lévy process that does not have negative jumps; this input process includes Brownian motion as a special case. However, exact asymptotic results there have not been fully proved yet (see Section 1 of [26] for some more discussions). For a two-dimensional reflecting random walk on the lattice with skip-free transitions, the book [8] (see also [16]) derived certain expressions for the generating function of the stationary distribution from a certain stationary equation that is analogous to our (2.3). Their techniques either use analytic extensions on Riemann surfaces or reduce the problem to the Riemann boundary value problem. These techniques may be useful to our problem in this paper, but we have not explored them here.
A recent paper [26] pioneered another analytic approach for a special case of SRBM. That SRBM arises from a similar tandem queue as in [10,19] but with an intermediate input, for which an explicit form is only partially available for the moment generating function of the stationary distribution. The authors first find the convergence domain of the moment generating function, namely, From the convergence domain, it is relatively easy to find the singularities of ϕ(zc) for each direction c. In this paper we take this analytic approach and show its full potential. As in [26], we consider some boundary moment generating functions that capture the reflections on the boundary faces. Unlike [26], we need to carefully study a relationship governing these moment generating functions. In this paper we do not assume any a priori information on the stationary distribution, whereas in [26] the marginal stationary distribution corresponding to the first node of the tandem queue is known. This forces us to seek a precise relationship among these moment generating functions (Lemma 4.1). This relationship is critical for us to characterize the convergence domain in Theorem 2.1.
Once the convergence domain is obtained, we employ analytic function theory to arrive at our main results, Theorems 2.2 and 2.3. Interestingly, it turns out that we can go beyond these results for some cases, obtaining a refinement of the exact asymptotics in (1.7). For example, the refinement can take the form where 0 < α c ≤ α d and b c , b d , κ c and κ d are some constants. We will briefly discuss this type of refinement in Section 8.
Determining exact asymptotics for two-dimensional SRBMs has been a difficult problem. Harrison and Williams [14] proved that when Σ and R satisfy the so-called skew symmetry condition, the stationary distribution of a d-dimensional SRBM has a product-form, each marginal being exponential. As a consequence, when the skew symmetry condition is satisfied, the tail asymptotic function f c (x) has the form x κ e −αcx , where κ takes one of the integers in 0, 1, . . . , d − 1. Dieker and Moriarty [4] proved that when d = 2 and a certain condition on Σ and R is satisfied, the two-dimensional stationary density is a finite sum of exponentials. Thus, the exact asymptotic in any direction c is known. For an SRBM arising from a tandem queue, Harrison [10] derives an explicit form for the two-dimensional stationary density. In this case, the exact asymptotic can also be computed; this is carried out in [19]. Except for these special cases and the one studied in [26], the exact asymptotics for two-dimensional SRBMs are not known. A part of the present results have recently been conjectured by Miyazawa and Kobayashi [25], which also includes conjectures for SRBMs in d ≥ 3 dimensions.
The analytic approach that is fully explored in this paper is general and should be applicable to discrete time reflecting random walks as well, as long as they are "skip free". When a random walk is not skip free, additional difficulties will show up. In such a case, the Markov additive approach (see, for example, [9]) will likely play a major role, although our analytic approach is still relevant; see [17,18].
Our results are closely related to the large deviations rate function I(v) for v ∈ R 2 + . The rate function I(v) is defined as a lower semi-continuous function that satisfies lim sup for any measurable B ⊂ R 2 + , where B and B o are closure and interior of B. When (1.11) and (1.12) are satisfied for a positive recurrent SRBM, it is said that the large deviations principle (LDP) holds with rate function I(v). The LDP is verified by Majewski [21,22] for an SRBM when R is an M matrix and R −1 µ < 0. When the latter two conditions are satisfied, the rate function I(v) is characterized as a solution to a variational problem. This M-matrix condition can be relaxed (see, e.g., [6]), but there is no LDP established in the literature when R is a completely-S matrix. Despite the lack of such an LDP, [1,11] studied the corresponding variational problem and derived an implicit characterization of its solution. Denoting as a consequence of our results, we have that the limit − lim exists and equals to the constant α c . Thus, we have verified the large deviations limit for B c , and the decay rate α c is also referred to as a rough asymptotic or logarithmic asymptotic. More discussion on LDP will be presented in Section 8.
In Section 2, we introduce various geometric objects that are associated with an SRBM. For an SRBM that has a stationary distribution, we classify it into one of the three categories, I, II and III, based on some properties of these geometric objects. The characterization of the convergence domain D is stated in Theorem 2.1. The domain has a geometric description that uses the fixed point equations (2.8) and (2.9). Theorem 2.2 states exact asymptotic results for SRBMs in Category I, and Theorem 2.3 states exact asymptotic results for SRBMs in Category II. The results and proofs for Category III are omitted because it is symmetric to Category II. Section 3 gives a constructive procedure to solve the fixed point equations. This procedure is critical for us to iteratively identify parts of the convergence domain. Section 4 studies a key relationship among moment generating functions. This relationship and the iterative procedure in Section 3 allow us to identify the extreme points of the convergence domain D in Section 5. Section 6 presents some complex analysis preliminaries to the proofs of our main results. Section 7 devotes to the proofs of Theorems 2.1-2.3. Section 8 presents some concluding remarks.
2. Geometric properties and the main results. In this paper we consider a two-dimensional SRBM Z with data (Σ, µ, R). Setting R = r 11 r 12 r 21 r 22 , throughout the paper except for Lemma 2.1, we assume that R is a P-matrix and (R, µ) satisfy (1.5); namely, A P-matrix is a completely-S matrix; see [2]. Thus, it follows from [29] that, under condition (2.1), the SRBM exists and is unique in distribution. Together, conditions (2.1) and (2.2) are necessary and sufficient for the twodimensional SRBM to have a stationary distribution [15,11]. When it exists, the stationary distribution is unique. As before, we use Z(∞) to denote a two-dimensional random vector that has the stationary distribution.
As discussed in Section 1, the convergence domain (1) of the two-dimensional moment generating function ϕ(θ) defined in (1.9) is of primary importance in determining the asymptotic tail of P{ c, Z(∞) ≥ u} as u → ∞. It turns out that the moment generating function ϕ(θ) is closely related to two boundary moment generating functions that we now define. For that, we first introduce two boundary measures. It follows from Proposition 3 of [3] that each component of E π (Y (1)) is finite, where E π (·) denotes the conditional expectation given that Z(0) follows the stationary distribution π. For a Borel set A ⊂ R 2 + , define Clearly, ν i defines a finite measure on R 2 + , which has a support on boundary {x ∈ R 2 + : x i = 0}. Let ϕ i be the moment generating function for ν i ; namely, An ellipse for µ1 < 0 and µ2 > 0: the ellipse γ(θ) = 0 intersects ray γ1(θ) = 0 at θ (2,r) and ray γ2(θ) = 0 at θ (1,r) . Its tangent at the origin is orthogonal to µ = (µ1, µ2). Condition (2.2) means that the angle formed by vector µ and ray γ k (θ) = 0 is more than π/2, k = 1, 2. The pair ϕ 1 and ϕ 2 are referred to as the boundary moment generating functions. We will prove the following facts in Proposition 4.1. For any θ = (θ 1 , θ 2 ) ∈ R 2 with ϕ(θ) < ∞, we have that ϕ 1 (θ 2 ) < ∞ and ϕ 1 (θ 2 ) < ∞. Furthermore, the following key relationship among moment generating functions holds: and R k is again the kth column of R. Now we define some geometric objects that will play an important role for us to fully explore the key relationship (2.3) to characterize the convergence domain D defined in (1). Because Σ is nonsingular, γ(θ) = 0 defines an ellipse ∂Γ that passes through the origin. We use Γ to denote the interior of the ellipse; namely, Define Γ to be the closure of Γ. From the definition of ∂Γ, µ = (µ 1 , µ 2 ) is orthogonal to the tangent of the ellipse at the origin; see Figure 1. It is clear that γ k (θ) = 0 is a line passing through the origin, k = 1, 2. For future purpose, we assign a direction for each line. For the line γ 1 (θ) = 0, the direction is (−r 21 , r 11 ). For the line γ 2 (θ) = 0, the direction is (r 22 , −r 12 ). Each directional line is called a ray. We use the terms line and ray interchangeably. Proof. Let v (1) = (−r 21 , r 11 ) and v (2) = (r 22 , −r 12 ). Then, tv (1) with variable t > 0 represents ray 1 with the 2nd coordinate to be positive, and tv (2) represents ray 2 with the 1st coordinate to be positive.
Since vector µ ≡ (µ 1 , µ 2 ) is orthogonal to the tangent of the ellipse γ(θ) = 0 at the origin and directed to the outside of ellipse, the conditions: are equivalent to that ray 1 intersects the ellipse at a point θ with θ 2 > 0 and ray 2 intersects the ellipse at a point θ with θ 1 > 0. Clearly, (2.4) is identical with (2.2). Thus, the first claim is proved.
Suppose both of µ 1 and µ 2 are nonnegative under (2.1) and (2.2). Then, both of r 12 and r 21 must be positive by (2.2). Multiplying the left and the right inequalities of (2.2) by r 11 and r 12 , respectively, and then adding them together, we have (r 22 r 11 − r 12 r 21 )µ 1 < 0.
But this is impossible because of (2.1). Hence, the second claim is proved.
We use θ (1,r) = 0 to denote the intersecting point of the ray γ 2 (θ) = 0 and the ellipse γ(θ) = 0, and similarly use θ (2,r) = 0 to denote the intersecting point of the ray γ 1 (θ) = 0 and the ellipse γ(θ) = 0. Here r is mnemonic for ray. The unconventional index scheme for θ (k,r) will be made clear in the next lemma: it derives from the fact that θ (1,r) is close to the θ 1 axis and θ (2,r) is close to the θ 2 axis.
Define open sets Clearly, they are nonempty. To abuse notation slightly, we define Then ∂Γ 1 is the portion of boundary ∂Γ that is below line γ 2 (θ) = 0. Similarly, ∂Γ 2 is the portion of boundary ∂Γ that is above line γ 1 (θ) = 0.
The following pair of fixed points (τ 1 , τ 2 ) plays a critical role in this paper: Setting Γ max = {θ ∈ R 2 : θ <θ for someθ such that γ(θ) > 0}, we have the following theorem characterizing of the convergence domain D.
For Categories II and III, τ ∈ ∂Γ must hold ( Figures 5-6). It is possible that τ ∈ Γ for Category I (see Figure 10 in Section 6 for an example). As a convention, when τ ∈ Γ, we set We now give the tail probability asymptotic of the convex combination for each direction vector c ∈ R 2 + . For a point x = 0 in R 2 , by "line x" we simply mean the line passing through the origin and x. Recall that α c is used to denote the exponent in the exact asymptotic (1.6). Following the discussion in Section 1 and (1), α c should be given by (2.13) α c = α > 0 such that αc ∈ ∂D ∩ R 2 + .
Indeed, throughout this paper, α c is defined through (2.13). To compute α c , the intersection of the line c with ∂D is important. In particular, it is helpful to see in which part of the boundary ∂D ∩ R 2 + this intersection is located. For this, let β k be the angle of line η (k) , measured counter clockwise starting from θ 1 axis, and let β be the angle of line c. Point c is said to be below line η (k) if β < β k , and above line η (k) if β > β k . To give an analytic expression for α c , let z c be the nonzero solution of γ(zc) = 0. Then, α c of (2.13) is given by (2.14) We first consider Category I. Recall the definition of η (1) and η (2) in (2.11) and (2.12). By Lemma 2.3, condition , which is further equivalent to θ (1,max) ∈ ∂Γ 1 .
(c) When c is above line η (2) , the case is symmetric to (a).
For Categories II and III, we only consider Category II because of their symmetry. In Category II, (c) When c is above line τ , i.e, β 1 < β ≤ π/2, Theorem 2.1 will be proved in Section 7.1. Theorem 2.2 will be proved in Section 7.2, and Theorem 2.3 will be proved in Section 7.3.

Solution to the fixed point equations. Lemma 2.3 in Section 2
is an important lemma that establishes the existence and uniqueness of the solution τ to the fixed point equations. In this section, we prove this lemma. We separate the proof into two lemmas that are given below.
The following lemma is similar to Lemma 3.4 of [17] (see also Corollary 4.1 of [24]), but a proof in our setting is simpler.
One can check that (3.2) and (3.3) are equivalent to the following We now use induction to show that For n = 1, clearly (3.6) holds. By the definition of τ 2 . Thus, (3.7) holds for n = 1. Suppose that (3.6) and (3.7) hold for n. We would like to show that (3.6) and (3.7) hold for n + 1. Because 0 ≤ τ . Thus, we have proved (3.6) for n + 1. Now, by the definition of τ , proving (3.7) for n + 1.
By (3.6), the two sequences {τ : n ≥ 0} are nondecreasing and bounded. Thus is well defined. We now prove that It suffices to prove that By the continuity of f 2 , there exists an N > 0 such that for n ≥ N , . Thus, we have τ 1 = τ * 1 . We can prove similarly that Therefore, we have proved that τ 1 and τ 2 satisfy (2.8) and (2.9). It remains to show that τ = (τ 1 , τ 2 ) = (0, 0). For this, it suffices to prove that τ (1) = (0, 0). Recall that Since vector (µ 1 , µ 2 ) is orthogonal to the tangent of the ellipse ∂Γ at the origin and at least one of µ 1 and µ 2 is negative because of Lemma 2.1, the ellipse ∂Γ intersects at least one of the two regions This implies that at least one of τ

4.
A key relationship among generating functions. In this section, we prove the key relationship (2.3) among the moment generating functions. The following lemma proves the relationship under several sets of conditions. This lemma is the key to the proofs of Theorems 2.1-2.3. Recall that Γ 1 and Γ 2 are defined in (2.7).

Lemma 4.1. Assume that conditions (2.1) and (2.2) hold. (a) For each
We will present the proof of this lemma later in this section. The tool for the proof is the basic adjoint relationship (4.1) below that governs the stationary distribution π and the corresponding boundary measures ν 1 and ν 2 . To state the basic adjoint relationship, let C 2 b (R 2 + ) be the set of twice continuously differentiable functions f on R 2 + such that f and its first-and second-order derivatives are bounded. , Then the basic adjoint relationship takes the following form: The basic adjoint relationship (4.1) is now standard in the SRBM literature. It was first proved in [13] for an SRBM in R d + for any integer d ≥ 1 when R is an M matrix. Extension to a general SRBM, when its stationary distribution exists, can be found, for example, in [3].
This f is generally not in C 2 b (R 2 + ), and thus (4.1) can not be applied directly. To overcome this difficulty, we construct a sequence of functions {f n } ⊂ C 2 b (R 2 + ) to approximate f . To this end, for each positive integer n, define function g n as Then, g n is continuously differentiable on R. We then define h n as Clearly, for each fixed n, h n (u) is twice continuously differentiable. Furthermore, for each u ∈ R + , h n (u) and g n (u) are monotone in n, and For each n, one can verify that −1 ≤ g ′ n (s) ≤ 0 for all s ∈ R + and f n ( Taking limit on both sides of (4.3) as n → ∞ and using monotone convergence theorem and (4.4), one has We first show that ϕ 2 (θ 1 ) < ∞. If θ 1 ≤ 0, the conclusion is trivial. Now we assume θ 1 > 0. Without loss of generality, we assume that From inequality (4.5), we have γ 2 (θ) < 0 and ϕ 1 (θ 2 ) < ∞. Letting n → ∞ in both sides of (4.3) as in the proof of part (a), we conclude the finiteness of ϕ 2 (θ 1 ). The proof for We would like to show that ϕ(θ) < ∞. For this, we again use (4.3). Applying the facts that This completes the proof since γ(θ) > 0.

Extreme points of the convergence domain.
This section is a preliminary to the proof of Theorem 2.1. In this section, we identify extreme points of the convergence domain D defined in (1). For this, we establish two lemmas, Lemmas 5.1 and 5.2. A consequence of Lemma 5.1 and part (c) of Lemma 4.1 is that Γ max . Lemma 5.2 shows a partial converse. We can not fully establish this converse in this section. To establish converse D ⊂ Γ (τ ) max , we need to use complex variable functions and their analytic extensions as discussed in Section 6.
Proof. We use Lemma 4.1 to iteratively expand a confirmed region on which ϕ(θ) < ∞. Recall two sequences {τ : n ≥ 0} defined in the proof of Lemma 3.2 in Section 3. We use induction to prove that, for each n ≥ 0, condition θ 1 < τ we can choose someθ 2 such that θ 2 <θ 2 < τ (n) 2 This completes the induction argument. Now the conclusion of part (a) follows from (3.8), Lemmas 3.1 and 3.2.
In the case when τ ∈ Γ (see Figure 10), Lemma 5.2 gives a complete converse. Therefore, Lemmas 5.1 and 5.2 give a complete proof of Theorem 2.1 when τ ∈ Γ; namely, In general, to establish the complete converse D ⊂ Γ (τ ) max , in addition to Lemma 5.2, we need to show that ϕ 2 (θ 1 ) = ∞ when θ 1 > τ 1 in the case of or τ 2 = f 2 (τ 1 ). We also need to verify that, if θ > τ and θ ∈ Γ max , then ϕ(θ) = ∞. These cases will be covered in Section 7.1 after the discussion of complex variable functions and their analytic extensions in the next section.
6. Analytic extensions and singular points. Recall that a complex function g(z) is said to be analytic at z 0 ∈ C if there exists a sequence {a n } ⊂ C and some ǫ > 0 such that ∞ n=0 a n (z − z 0 ) n is absolutely convergent and is equal to g(z) for each z ∈ C with |z − z 0 | < ǫ. A point z ∈ C is said to be a singular point of a (complex variable) function if the function is not analytic at z. A moment generating function for a nonnegative random variable can be considered as a function of complex variable z ∈ C that is analytic, at least when the real part ℜz of z is negative. We use complex variable functions to identify the convergence domain D through their singular points. To this end, the following lemma is useful although it is elementary. The lemma corresponds with Pringsheim's theorem for a generating function (e.g., see Theorem 17.13 in Volume 1 of [23]). Lemma 6.1. Let g(λ) = ∞ 0 e λx dF (x) be the moment generating function of a probability distribution F on R + with real variable λ. Define the convergence parameter of g as Then, the complex variable function g(z) is analytic on {z ∈ C; ℜz < c p (g)} and is singular at z = c p (g).
Remark 6.1. For a complex function g : C → C that is singular at z = z 0 ∈ C, g(z 0 ) may be finite at z 0 , but Lemma 6.1 shows that, if g(z) is a moment generating function, then g(x) = ∞ for x ∈ (ℜz 0 , ∞).
The following corollary is immediate from this lemma.
Corollary 6.1. Under the same notation of Lemma 6.1, we have the following two facts.
From Lemmas 5.1 and 6.1 and (c) of Lemma 4.1, we have the following lemma.
The remaining of this section is to prove that ϕ k (z) is singular at z = τ k and to determine the nature of singularity at z = τ k , k = 1, 2. For that, we are going to relate ϕ 1 (z) and ϕ 2 (z) through (2.3). This motivates us to study the roots of γ(θ) = 0. For that, it is convenient to have function representations for different segments of the ellipse ∂Γ, which is defined by the equation γ(θ 1 , θ 2 ) = 0 for (θ 1 , θ 2 ) ∈ R 2 . Recall that f 2 (θ 1 ) represents the lower half of the ellipse ∂Γ and f 1 (θ 2 ) represents the left half of the ellipse ∂Γ. As it will be clear from (6.9) and (6.8) below that f 1 and f 2 play key roles for finding the singularities of ϕ 1 (z) and ϕ 2 (z). Recall again that θ (1,max) is the right-most point on ∂Γ and θ (2,max) is the highest point on ] is well defined, and is given by formula ]. A formula for f 1 (θ 2 ) can be written down similarly.
The following lemma says that f 2 (z) has an analytic extension. Recall that, for a multi-valued function f (z) on C, a point z 0 ∈ C is said to be a branch point of f if there exists a neighborhood such that an arbitrary closed continuous curve around z 0 in the neighborhood carries each branch of f to another branch of f (see Section 11 in Volume I of [23] for its precise definition). For example, f (z) = (z −z 0 ) 1/n with integer n ≥ 2 is an n-valued function with branch point z 0 . The integer n is referred to as the multiplicity of the branch point. Obviously, each branch of this function is analytic on and arg z ∈ (−π, π) is the principal part of the argument of a complex number z.
Since f 2 (z) is multivalued for complex number z ∈ C, we take its branch ). We use the same notation f 2 (z) for this branch throughout the paper. , each with multiplicity two. Furthermore, Proof. Because θ are two roots of the quadratic equation (µ 2 + Σ 12 η 1 ) 2 − Σ 22 (Σ 11 η 2 1 + 2µ 1 η 1 ) = 0, the quadratic function must be equal to where det(Σ) = Σ 11 Σ 22 − Σ 2 12 . Thus the complex variable version of f 2 in (6.2) must have the following representation: Then, from our choice of the branch of f 2 , g(z) can be written as where ω + (z) = arg(z − θ as claimed. Note that ω − (z), ω + (z) ∈ (−π, π) and their signs are different if z is not on the real line. Hence, − π 2 < ω − (z)+ω + (z) 2 < π 2 . This and the fact that We now prove (6.2). For this, it suffices to prove that for any z ∈ C with ℜz ∈ (θ , so we have cos ω + (z) ≥ 0 and cos ω − (z) ≥ 0. Hence, This indeed holds because It remains to prove (6.3). From (6.4) and (6.5), we have . Let z ∈ C satisfy ℜz ≥ 0 and z is not real. Then, z can be expressed as z = re iω 0 with r > 0 and Clearly, r < s. Since z and z + θ 1 have the same imaginary component, ω 0 and ω 1 have the same sign. Hence, it is sufficient to consider for ω 0 , ω 1 > 0 to prove the right side of (6.6) to be less than zero. In this case, it is easy to see that arg(−z) = ω 0 − π and ω 1 < ω 0 . Using these notation, it follows from (6.6) that If Σ 12 ≥ 0, then the last term of this inequality is negative, and therefore (6.6) is negative. Hence, (6.3) holds for δ = 0. Otherwise, we choose δ ∈ (0, π 2 ) such that which is possible because its left side goes to infinity as δ ↑ π 2 . Then, for ω 0 > δ, we have, by the monotonicity of sine and cosine functions on the interval [0, π 2 ], This implies that the right side of (6.7) is negative. Thus, combining this result with (6.2), we obtain (6.3). This completes the proof.
Similarly, the following lemma classify the singularity of ϕ 1 (z) at z = τ 2 when τ 2 < θ (2,max) 2 . Its proof is analogous to the proof of Lemma 6.6, and is omitted.
Here, ϕ ′ 1 (θ 2 ) is the derivative of ϕ 1 (θ 2 ) and we have used the fact that To prove (6.24) and (6.25), we first note that ϕ 1 (θ , it is easy to see from the definition of f 2 that, for z ∈ G δ (θ We also note that ).
7. Proofs of the theorems. In this section, we prove Theorems 2.1, 2.2 and 2.3. The ideas of proofs have the same spirit as those in [26] (see also [17]), but we need more ideas. The proofs use the analytic extensions of complex variable functions developed in Section 6, complex inversion techniques in Appendix C, and decompositions of moment generating functions in Appendix D.
7.1. The proof of Theorem 2.1 . The next lemma partially characterizes the boundary of the domain D.
We first note that, if τ ∈ Γ, there is no θ such that θ < τ and θ ∈ Γ max . Hence, we can assume that τ ∈ Γ. We then must be in Category I and η 1 )} be the segment of the ellipse ∂Γ that is within both the upper half and the right half of the ellipse and is below τ . In Figure 12, ∂Γ + is the piece of the ellipse that borders the shaded region. To prove that claim, it suffices to prove that for each relatively open neighborhood B ⊂ ∂Γ + , there exists a θ ∈ B such that ϕ(θ) = ∞ becauseθ ≥ θ implies ϕ(θ) = ∞.

The proof of Theorem 2.2.
We now start to prove Theorem 2.2. From (1.9), the definition of α c in (2.13), Lemma 6.1, and Theorem 2.1, we know that the moment generating function ψ c (z) for c 1 Z 1 + c 2 Z 2 is analytic for ℜz < α c and is singular at α c . Again from Theorem 2.1, α c ≤ min(τ 1 /c 1 , τ 2 /c 2 ).
Recall that z c is the nonzero root of γ(zc) = 0. Let ζ c (z) = −( 1 2 c, Σc z + c, µ ). Then, z c is the root of ζ c (z) = 0, and Therefore, for ℜz < α c and z = z c , where it can be checked that (7.3) is valid for z = 0. The numerator in the right side of (7.3) is analytic for ℜz < min(τ 1 /c 1 , τ 2 /c 2 ) because of Lemma 6.2. Because ψ c (z) is analytic for ℜz < α c , z = z c must be a removable singularity of the right side of (7.3) if z c < α c . Hence, if z c < α c , we have If z c ≥ α c , from (7.3), it is easy to see that the moment generating function ψ c (z) can be expressed as a linear combination of the form in (D.2) with k = 1. If z c < α c , from (7.4), ψ c (z) can be expressed as a linear combination of the form in (D.3). By Lemmas D.2 and D.3, each term in these two linear decompositions is the moment generating function of a continuous density since ϕ 1 and ϕ 2 are the moment generating functions of the measures ν 1 and ν 2 , respectively, and both ν 1 and ν 2 have densities (see [3,13]). Thus, by Lemma D.1, the distribution of c, Z(∞) also has a continuous density p c (x) on [0, ∞).
In the following proof, for many cases we actually prove that the density p c (x) has an exact asymptotic, which implies that the exact asymptotic for the tail distribution q c (x) = P( c, Z(∞) ≥ x), x ≥ 0 by Lemma D.5. However, in some cases, we are not able to establish the exact asymptotic for density p c (x). In these cases, we work with the moment generating function ψ c (z) of the tail probability q c (x) directly, where One can check that for ℜz < α c (7.5) and z = 0 is a removable singularity of ψ c (z).
Note that when α c c = θ (1,r) , we are unable to verify condition (7.10) for ψ c (z), and hence not able to obtain the exact asymptotic for p c (x).
(a.ii) Assume that τ 1 = θ (1,max) 1 . In this case we apply Lemma C.2 in addition to Lemma D.4 to get exact asymptotics. Now we examine the details. Because τ 1 = θ (1,max) 1 , we have z c < α c and hence ζ c (α c ) < 0. Thus, we can use the expression (7.4) with functions h 1 and h 2 of (7.2). Because ϕ 1 (c 2 z) is analytic at z = α c , h 1 (α c + ǫ) is finite, for some ǫ > 0. By Lemma D.3, h 1 is a moment generating function of a signed measure ξ 1 . Because h 1 (α c + ǫ) is finite, the tail of ξ 1 decays as On the other hand, the function h 2 is singular at z = α c . The analytical behavior h 2 (z) around z = α c is identical to that of ϕ 2 (c 1 z). We apply Lemma 6.8 to find the analytical behavior of ϕ 2 (c 1 z) at α c . We consider two separate cases.
We next consider the case when η (1) = θ (1,r) . In this case, by Lemma 6.6, ϕ 2 (z) has a double pole at z = τ 1 , thus ϕ 2 (c 1 z) has a double pole at z = α c . Using the representation (7.9), we see that ψ c (z) has a double pole at z = α c . One can check, just as in the proof of case (a.i) in Theorem 2.2, that conditions (C1a), (C1b) and (Cbc) are satisfied for ψ c (z) with m = 1 and k 0 = 2. By Lemma C.1, q c (x) has exact asymptotic that is given by (7.15).
8. Concluding remarks. In this section, we briefly discuss four topics. They are the non-singular assumption on the covariance matrix Σ, the large deviations rate function, the fine exact asymptotics, and an extension to SRBMs in more than two dimensions.
Singular covariance matrix. Throughout this paper, we have assumed that Σ is positive definite, which is equivalent to be non-singular. In applications, the covariance matrix Σ may be singular. Consider, for example, a tandem queue that has a single Poisson arrival process at station 1. Each station has a single server, and the service times at each station are deterministic. In this case, the two-dimensional queue length process can be modeled by an SRBM that has a singular covariance matrix [12]. When Σ is singular, the ellipse ∂Γ becomes a parabolic curve and the set Γ is not bounded. The exact asymptotics analysis should be analogous to the analysis in this paper, but the detailed analysis is not attempted in this paper.
Large deviations. We next consider the connection of convergence domain D with large deviations. As we discussed in Section 1, the rough asymptotics of P(Z(∞) ∈ uB) may be interesting for B ∈ B(R 2 + ). We here consider the large deviations rate function I in (1.11) and (1.12). Let B be any convex, closed set of R 2 + and u be a positive number. Since Z(∞) ∈ uB implies u inf v∈B v, θ ≤ θ, Z(∞) for each θ ∈ R 2 , we have where inf v∈B v, θ may be −∞, for which the above inequality is trivial. Hence, as long as ϕ(θ) < ∞, that is, θ ∈ D, Furthermore, from the fact that θ, v for θ ∈ D attains the supremum on the closure of D min for any v ∈ R 2 + (see Figure 13), the right side of the above inequality equals inf v∈B sup θ∈D v, θ . Hence, (8.1) yields lim sup Thus, if the rate function I exists, then, from (1.12), we have Furthermore, comparing (8.2) with the rate function I in (1.11), we can say this J is a very good candidate for the rate function I. One may ask whether J is indeed the rate function using the I obtained in [1,11]. This question is also related to how the optimal path in the sample path large deviations is related to the present results. These questions will be answered in a subsequent paper. In particular, it will be shown that the equality in (8.3) holds if and only if τ ∈ D.
Another interesting question related to the large deviations is the exact asymptotics of P(Z(∞) ∈ xv + B) for each directional vector v ≥ 0 and a closed subset B of R 2 + . This is a harder problem. It will also be investigated in the subsequent paper.
Fine exact asymptotics. For each direction c ∈ R 2 + , we have derived the exact tail asymptotics for tail probability q c (x) in the form x κ e −αx through the left-most singular point of ψ c . If we carefully examine our proofs in Section 7.2, we can find its second left-most singular point in many cases. This suggests a possibility to obtain a finer asymptotic function than the one given in (1.6). That is, we may refine the tail asymptotics in the form of (1.10). However, it is notable that constants b c , b d there can not be obtained in general. Here, we present fine asymptotics f c (x) for q c (x) as x → ∞ for three cases. They illustrate the potential and difficulty for pursuing general cases.
(i) c is strictly above η (1) and below or on line τ : (see case (b) of Theorem 2.2): We first assume that c is strictly below line τ . The left-most singular point α c of ψ c (z) is the root of ζ c (z) by (7.3). Hence, ψ c (z) has a simple pole at z = z c . The next singularity of ψ c (z) occurs when either ϕ 1 (c 2 z) or ϕ 2 (c 1 z) is singular at z. Because c is below line τ , we have τ 1 c 1 < τ 2 c 2 . Then, ϕ 2 (c 1 z) is singular at point z = τ 1 c 1 , which is smaller than τ 2 c 2 , the singular point of ϕ 1 (c 2 z). Therefore, z = τ 1 c 1 is the second left-most singular point of ψ c (z). We decompose ψ c of (7.3) as ψ c (z) = g 0 (z) + g 1 (z) where g 0 (z) = 1 ζ c (z) (γ 2 (c)ϕ 2 (c 1 α c ) + γ 1 (c)ϕ 1 (c 2 z)) , It is easy to see that g 0 (z) has a simple pole at z = α c , and is analytic for ℜz < τ 2 c 2 except for this pole. On the other hand, g 1 (z) has a removable pole at z = α c , and is analytically extendable for ℜz < τ 1 c 1 , and singular at z = τ 1 c 1 . In this case, ϕ 2 (c 1 z) has a simple pole there by Lemma 6.6 because of the assumption τ 1 < θ (1,max) 1 . Hence, by applying Lemma D.4 for g 0 and Lemma C.1 for g 1 , we have where b c and b d are positive constants.
subsets of Ω, and {F t } ≡ {F t , t ≥ 0} is an increasing family of sub-σ-fields of F, i.e., a filtration. Definition A.1 gives the so-called weak formulation of an SRBM. It is a standard definition adopted in the literature; see, for example, [7] and [30]. Note that condition (A.1) is equivalent to the condition that, for each t > 0, [27] showed that a necessary condition for a (R d + , µ, Σ, R)-SRBM to exist is that the reflection matrix R is completely-S (this term was defined in Section 1). [29] showed that when R is completely-S, a (R d + , µ, Σ, R)-SRBM Z exists and Z is unique in law under P x for each x ∈ S. Furthermore, Z, together with the family of probability measures {P x , x ∈ R d + }, is a Feller continuous strong Markov process.

APPENDIX C: COMPLEX INVERSION TECHNIQUE
As we planed in Section 6, we use complex variable moment generating functions for proving Theorems 2.2 and 2.3. In this section, we first present classical results for obtaining exact tail asymptotics from a complex variable moment generating function. For this, we refer to Doetsch [5] similarly to [26].
Let f be a nonnegative valued, continuous and integrable function on [0, ∞), and define the complex variable function g as where α 0 = c p (g)(≡ sup{θ ≥ 0 : g(θ) < ∞}). By Lemma 6.1, α 0 is the leftmost singular point of g(z), and g(z) is analytic for ℜz < α 0 . We analytically expand this function g, which is also denoted by g. We are interested in the following two cases for α 0 > 0.
We can similarly prove the following lemma, which is a special case of Theorem 35.1 in [5].
Lemma C.1. Let g be the moment generating function of a nonnegative, continuous and integrable function f . If the following conditions are satisfied for some integer m ≥ 1 and some number α m > 0, (C1a) there is a complex variable function g 0 (z) such that, some integers k j and some numbers c j , α j for j = 0, 1, . . . m − 1 such that 0 < α 0 < α 1 < · · · < α m−1 < α m and and g(z) − g 0 (z) is analytic for ℜz < α m , (C1b) g(z) uniformly converges to 0 as z → ∞ for 0 ≤ ℜz ≤ α m , (C1c) the integral For case (C2), the situation is a bit complicated, but the idea is essentially the same. We need the counter integral along the boundary of G δ (α 0 ) instead of Fourier inversion formula. The following lemma is a special case of Theorem 37.1 of Doetsch [5].

APPENDIX D: DECOMPOSING A MOMENT GENERATING FUNCTION
In application of Lemmas C.1 and C.2, we need to verify that the density function f is continuous in (0, ∞). Furthermore, we may need to decompose the moment generating function g of interest into the linear combination of moment generating functions for which those lemmas are applicable. We present their details in this appendix. For convenience, we refer to an integrable nonnegative function as a density. We first note the following basic fact. In verifying the continuity of densities as well as decomposing moment generating functions, the following two results are useful.

2)
Then h is the moment generating function of some density function on [0, ∞). In particular, this density is continuous if ξ has has no atom.
Proof. Let g 1 (s) = (s 0 /(s 0 − s)) k for s ≤ 0. Then g 1 (s) is the moment generating function of the Erlang distribution with order k and mean (ks 0 ) −1 . Clearly, h(s) = g 1 (s)g 2 (s) for s ≤ 0, where g 2 (s) = g(s)/s k 0 is the moment generating function of measure ξ/s k 0 . Therefore, h is the moment generating function of the convolution of measure ξ/s k 0 and the Erlang distribution. Since the Erlang distribution has a density, this convolution must have a density. The remaining statement is immediate from this convolution and the fact that ξ has no atom.
Lemma D.3. Let g be the same as in Lemma D.2, and assume g(s 1 ) is finite for some constant s 1 ∈ R. Let h(s) = g(s 1 ) − g(s) s 1 − s , s < s 1 .

(D.3)
Then h is also the moment generating function of a density function on [0, ∞), and has the same convergence parameter as g (see Lemma 6.1 for its definition). In particular, this density is continuous if ξ has no atom.
Proof. The statement on the convergence parameter is immediate from the definition of h. Hence, h(s) is the moment generating function of the density e −s 1 x ∞ x e s 1 u × ξ(du), which is continuous in x if ξ has no atom.
In application of Lemma C.1, the condition (C1b) is annoying. In our application, the following result is sufficient.
Lemma D.4. Assume a complex variable function h is given by (D. 2) with g such that ξ has no atom. Denote a continuous density on [0, ∞) whose moment generating function is h byf . If g(s 0 + ǫ) < ∞ for some ǫ > 0, theñ Proof. For k = 1, we decompose g as