3.1. Classes and chains of classes

We will require some concepts from combinatorial spectral theory. Much of this subsection is adapted from [Reference Rothblum32]. Similar treatments can be found in [Reference Berman and Plemmons7] and [Reference Tam37].

Let A be a square matrix with directed graph $G(A)=(V(A),E(A))$ . We say that vertex $m\in V(A)$ has access to vertex $n\in V(A)$ if $m=n$ or if there exists a path of directed edges in E(A) from m to n. We say that vertices $m,n\in V(A)$ communicate if they have access to one another. Communication defines an equivalence relation on V(A), and thus partitions V(A) into disjoint equivalence classes $\gamma_1,\dots,\gamma_M\subseteq V(A)$ . Any two vertices within the same equivalence class communicate. The sets of vertices $\gamma_1,\dots,\gamma_M$ are called the classes of A. They are the vertex sets of the strongly connected components of G(A). The matrix A is irreducible if and only if V(A) is the single class for A, i.e. $M=1$ .

If $\gamma$ and $\delta$ are classes of A, then we say that $\gamma$ has access to $\delta$ if there exists a vertex $m\in\gamma$ with access to some vertex $n\in\delta$ . In this case we write $\gamma\preceq\delta$ . If $\gamma\preceq\delta$ and $\gamma\neq\delta$ then we write $\gamma\prec\delta$ . The binary relation $\preceq$ defines a partial order on the set of all classes of A, while $\prec$ defines a strict partial order. If $\gamma$ is a class of A such that $\delta\nprec\gamma$ for every class $\delta$ of A, then $\gamma$ is said to be initial. If $\gamma$ is a class of A such that $\gamma\nprec\delta$ for every class $\delta$ of A, then $\gamma$ is said to be final. Every square matrix has at least one initial class and at least one final class.

To illustrate the preceding concepts we consider the $10\times10$ Metzler matrix defined by

(5) \begin{equation}A=\begin{bmatrix}-3& \quad 0& \quad 0& \quad 1& \quad 0& \quad 3& \quad 0& \quad 0& \quad 0& \quad 0\\0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 4\\0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 0\\12& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 4& \quad 0& \quad 0& \quad 0\\0& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0\\0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 1& \quad 0& \quad 7& \quad 0\\0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 1& \quad 0& \quad 0& \quad 0& \quad 0\\0& \quad 0& \quad 0& \quad 0& \quad 5& \quad 0& \quad 0& \quad -3& \quad 0& \quad 0\\0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 3& \quad 0\\0& \quad 2& \quad 0& \quad 0& \quad 0& \quad 0& \quad 0& \quad 2& \quad 0& \quad -1\end{bmatrix}\!.\end{equation}

The pattern of zero and nonzero off-diagonal entries of A generates the directed graph G(A) displayed in Figure 1. There are exactly five classes of A, which we label by

\begin{equation*}\gamma_1=\{1,4\},\quad\gamma_2=\{3,6,7\},\quad\gamma_3=\{2,10\},\quad\gamma_4=\{9\},\quad\gamma_5=\{5,8\}.\end{equation*}

The five classes of A satisfy the four strict partial orderings

\begin{equation*}\gamma_1\prec\gamma_2,\quad\gamma_2\prec\gamma_3,\quad\gamma_2\prec\gamma_4,\quad\gamma_3\prec\gamma_5,\end{equation*}

as well as the other strict partial orderings implied by those four via transitivity. Note however that $\gamma_4$ is incomparable to $\gamma_3$ and to $\gamma_5$ . In Figure 1, $\gamma_4$ is located horizontally between $\gamma_3$ and $\gamma_5$ , but we may equally well have located it to the left or right of both. There is a unique initial class of A, this being $\gamma_1$ , and exactly two final classes of A, these being $\gamma_4$ and $\gamma_5$ .

Figure 1.

The directed graph G(A) for the matrix A in (5).

It is always possible to relabel the rows and columns of a square matrix A in such a way that the resulting matrix is block upper triangular with blocks corresponding to pairs of classes of A. Specifically, given a square matrix A whose classes are $\gamma_1,\dots,\gamma_M$ , we may define

(6) \begin{align}\tilde{A}=\begin{bmatrix}A_{\gamma_1,\gamma_1}& \quad A_{\gamma_1,\gamma_2}& \quad \cdots& \quad A_{\gamma_1,\gamma_M}\\[3pt]A_{\gamma_2,\gamma_1}& \quad A_{\gamma_2,\gamma_2}& \quad \cdots& \quad A_{\gamma_2,\gamma_M}\\[3pt]\vdots& \quad \vdots& \quad \ddots& \quad \vdots\\[3pt]A_{\gamma_M,\gamma_1}& \quad A_{\gamma_M,\gamma_2}& \quad \cdots& \quad A_{\gamma_M,\gamma_M}\end{bmatrix}\!,\end{align}

where $A_{\gamma_j,\gamma_k}$ is the matrix obtained by deleting those rows of A whose indices do not belong to $\gamma_j$ and those columns of A whose indices do not belong to $\gamma_k$ . If the notation $\gamma_1,\dots,\gamma_M$ is assigned to the classes of A in such a way that $\gamma_k\nprec\gamma_j$ whenever $j<k$ , then the blocks in (6) below the diagonal are guaranteed to be zero. This is always possible because $\prec$ is a strict partial order. The diagonal blocks in (6) are irreducible because all pairs of vertices in a class communicate. The construction of $\tilde{A}$ is equivalent to applying a permutation to the rows and columns of A. Thus A and $\tilde{A}$ are similar and share many spectral properties.

In our example, we may transform A to the form (6) by applying the permutation

\begin{equation*}(1,2,3,4,5,6,7,8,9,10)\mapsto(1,6,3,2,9,4,5,10,8,7)\end{equation*}

to the rows and columns of A. This produces the matrix

which is block upper triangular with irreducible diagonal blocks.

In cases where A is Metzler it can be useful to distinguish between basic and nonbasic classes of A. Let A be a Metzler matrix with classes $\gamma_1,\dots,\gamma_M$ . Since A is similar to the block matrix $\tilde{A}$ obtained by permuting rows and columns as in (6), the spectral abscissae of A and $\tilde{A}$ are equal. A suitable assignment of the notation $\gamma_1,\dots,\gamma_M$ to the classes of A guarantees that $\tilde{A}$ is block upper triangular, so the spectral abscissa of $\tilde{A}$ is the maximum of the spectral abscissae of its diagonal blocks $A_{\gamma_k,\gamma_k}$ . Thus

\begin{equation*}\zeta(A)=\max_{k\in\{1,\dots,M\}}\zeta(A_{\gamma_k,\gamma_k}).\end{equation*}

We say that the class $\gamma_k$ is basic if $\zeta(A_{\gamma_k,\gamma_k})=\zeta(A)$ . Otherwise we say that $\gamma_k$ is nonbasic. Every Metzler matrix has at least one basic class.

In our example, straightforward calculations show that

\begin{align*}\zeta\begin{pmatrix}-3& \quad 1\\[2pt]12& \quad 1\end{pmatrix}=3,\quad \zeta\begin{pmatrix}0& \quad 0& \quad 1\\[2pt]0& \quad 0& \quad 1\\[2pt]1& \quad 1& \quad 0\end{pmatrix}=\sqrt{2},\quad \zeta\begin{pmatrix}1& \quad 4\\[2pt]2& \quad -1\end{pmatrix}=3,\quad \zeta\begin{pmatrix}1& \quad 1\\[2pt]5& \quad -3\end{pmatrix}=2.\end{align*}

Thus the basic classes of A are $\gamma_1$ , $\gamma_3$ , and $\gamma_4$ , and the nonbasic classes are $\gamma_2$ and $\gamma_5$ .

We will require the concept of a chain of classes. Given a square matrix A with classes $\gamma_1,\dots,\gamma_M$ , an $\ell$ -tuple of classes $(\gamma_{i_1},\dots,\gamma_{i_\ell})$ is called a chain from $\gamma_{i_1}$ to $\gamma_{i_\ell}$ , or simply a chain, if $\ell=1$ or if $\gamma_{i_k}\prec\gamma_{i_{k+1}}$ for each $k\in\{1,\dots,\ell-1\}$ . In our example, valid chains include, for instance,

(7) \begin{equation}(\gamma_1,\gamma_2,\gamma_4),\quad(\gamma_2,\gamma_5),\quad(\gamma_1),\quad(\gamma_1,\gamma_2,\gamma_3,\gamma_5),\quad(\gamma_1,\gamma_3),\quad(\gamma_1,\gamma_4).\end{equation}

Tuples of classes which are not valid chains include, for instance,

\begin{equation*}(\gamma_3,\gamma_4),\quad(\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5),\quad(\gamma_5,\gamma_2),\quad(\gamma_1,\gamma_1),\quad(\gamma_4,\gamma_5),\quad(\gamma_1,\gamma_3,\gamma_4).\end{equation*}

If A is Metzler then to each chain of classes of A we assign a nonnegative integer called its length. The length of a chain is the number of basic classes it contains. In our example, the chains listed in (7) have lengths 2, 0, 1, 2, 2, and 2, respectively.

In Theorem 2, to be stated in Section 3.2, and also in the Rothblum index theorem, to be stated in Section 3.3, a key role is played by the length of the longest chain of classes, by which we mean the maximum length of all chains. In our example there are exactly three basic classes, these being $\gamma_1$ , $\gamma_3$ , and $\gamma_4$ . The length of the longest chain of classes is 2. There is no chain of classes of length 3 because $\gamma_3$ and $\gamma_4$ cannot both be included in a single chain due to the fact that $\gamma_3\nprec\gamma_4$ and $\gamma_4\nprec\gamma_3$ .

We require one further concept related to chains, to be used only in the proof of Theorem 2. A chain of classes $(\gamma_{i_1},\dots,\gamma_{i_\ell})$ will be called a direct chain if $\ell=1$ or if, for each $k\in\{1,\dots,\ell-1\}$ , there exist vertices $m\in\gamma_{i_k}$ and $n\in\gamma_{i_{k+1}}$ such that (m, n) is a directed edge in E(A). If A is Metzler then the latter requirement is equivalent to requiring that $A_{\gamma_{i_k},\gamma_{i_{k+1}}}$ is semipositive for each $k\in\{1,\dots,\ell-1\}$ . In our example, valid direct chains include, for instance,

\begin{equation*}(\gamma_1,\gamma_2,\gamma_4),\quad(\gamma_1),\quad(\gamma_1,\gamma_2,\gamma_3,\gamma_5).\end{equation*}

Chains which are not direct chains include, for instance,

\begin{equation*}\quad(\gamma_2,\gamma_5),\quad(\gamma_1,\gamma_3),\quad(\gamma_1,\gamma_4).\end{equation*}

Every chain of classes is nested within some direct chain of classes. For instance, in our example, the chain $(\gamma_1,\gamma_4)$ is not a direct chain but is nested within the direct chain $(\gamma_1,\gamma_2,\gamma_4)$ . Both of these chains have length 2.

3.2. Main result

Our primary technical contribution is the following modification of Proposition 1 applicable in settings where $A(\alpha)$ need not be irreducible. It will be used in Section 4, in conjunction with Theorem 1, to show that the tail probabilities generated by two models of random growth resemble those of an Erlang distribution.

In condition (v) the notation $A_{\gamma,\delta}(z)$ refers to the matrix obtained by deleting the rows and columns of A(z) whose indices do not belong to $\gamma$ and $\delta$ respectively. This condition prevents the formation of connections to previously inaccessible classes as z moves away from $\alpha$ . Examples provided at the end of this subsection illustrate what can go wrong if conditions (v) or (vi) are not satisfied.

We will appeal to two lemmas on matrix algebra in our proof of Theorem 2. The first is contained in Theorems 2.3 (see condition $\text{N}_{38}$ ) and 2.7 in [Reference Berman and Plemmons7, Chapter 6].

The second lemma we require extends the well-known inverse formula for $2\times2$ upper triangular block matrices with square diagonal blocks, that is,

\begin{equation*}\begin{bmatrix}A_{1,1}& \quad A_{1,2}\\[2pt]0& \quad A_{2,2}\end{bmatrix}^{-1}=\begin{bmatrix}A_{1,1}^{-1}& \quad -A_{1,1}^{-1}A_{1,2}A_{2,2}^{-1}\\[2pt]0& \quad A_{2,2}^{-1}\end{bmatrix}\!,\end{equation*}

to the $N\times N$ case. We omit the proof as it is a straightforward application of induction.

Lemma 5. Let A be a nonsingular block matrix with (j, k)th block denoted by $A_{j,\, k}$ . Assume that A is block upper triangular with square diagonal blocks. Let $B=A^{-1}$ , and partition B into blocks $B_{j,\, k}$ conformably with the blocks of A. Then B is block upper triangular and for $j\leq k$  we have

(8) \begin{align}B_{j,\, k}=\sum({-}1)^{\ell-1}A_{i_1,i_1}^{-1}A_{i_1,i_2}A^{-1}_{i_2,i_2}\cdots A_{i_{\ell-1},i_\ell}A^{-1}_{i_\ell,i_\ell},\end{align}

where the sum runs over all integer tuples $(i_1,\dots,i_\ell)$ with $\ell\in\{1,\dots,k-j+1\}$ and $j=i_1 < \cdots < i_\ell=k$ .

Now we use Proposition 1 and Lemmas 4 and 5 to prove Theorem 2.

Proof of Theorem 2. Let M be the number of classes of $A(\alpha)$ , so that $M=1$ if $A(\alpha)$ is irreducible and $M\geq2$ if $A(\alpha)$ is reducible. Denote the classes of $A(\alpha)$ by $\gamma_1,\dots,\gamma_M$ . In view of the discussion in Section 3.1 we may assume without loss of generality that $A(\alpha)$ is an $M\times M$ upper triangular block matrix with irreducible diagonal blocks. Under condition (v) the matrix A(z) is then an $M\times M$ upper triangular block matrix with irreducible diagonal blocks for all $z\in\Omega$ , and we write

\begin{align*}A(z)& \quad =\begin{bmatrix}A_{1,1}(z)& \quad A_{1,2}(z)& \quad \cdots& \quad A_{1,M}(z)\\[3pt]0& \quad A_{2,2}(z)& \quad \cdots& \quad A_{2,M}(z)\\[3pt]\vdots& \quad \vdots& \quad \ddots& \quad \vdots\\[3pt]0& \quad 0& \quad \cdots& \quad A_{M,M}(z)\end{bmatrix}\!.\end{align*}

Lemma 5 implies that when A(z) is nonsingular its inverse is an $M\times M$ upper triangular block matrix of the form

\begin{align*}A(z)^{-1}& \quad =\begin{bmatrix}B_{1,1}(z)& \quad B_{1,2}(z)& \quad \cdots& \quad B_{1,M}(z)\\[3pt]0& \quad B_{2,2}(z)& \quad \cdots& \quad B_{2,M}(z)\\[3pt]\vdots& \quad \vdots& \quad \ddots& \quad \vdots\\[3pt]0& \quad 0& \quad \cdots& \quad B_{M,M}(z)\end{bmatrix}\!,\end{align*}

where the blocks $B_{j,\, k}(z)$ on and above the diagonal are given by (8). In particular, the diagonal blocks satisfy $B_{k,k}(z)=A_{k,k}(z)^{-1}$ for all z such that A(z) is nonsingular.

Each block $B_{j,\, k}(z)$ should be understood to be a holomorphic function defined on the set obtained by removing from $\Omega$ the discrete set of points at which A(z) is singular. Since $A(z)^{-1}$ is meromorphic on $\Omega$ , each block $B_{j,\, k}(z)$ is meromorphic on $\Omega$ , with a pole or removable singularity at each point at which A(z) is singular. To determine the order of the pole of $A(z)^{-1}$ at $\alpha$ it suffices to identify the blocks $B_{j,\, k}(z)$ which have a pole at $\alpha$ and find the maximum order of the pole among those blocks.

We begin by studying the behavior of the inverse submatrices $A_{k,k}(z)^{-1}$ . Since $\zeta(A(\alpha))=0$ , for each k we have $\zeta(A_{k,k}(\alpha))=0$ if $\gamma_k$ is basic and $\zeta(A_{k,k}(\alpha))<0$ if $\gamma_k$ is nonbasic. If $\zeta(A_{k,k}(\alpha))<0$ then $A_{k,k}(\alpha)$ is nonsingular and so $\alpha$ cannot be a pole of $A_{k,k}(z)^{-1}$ . Therefore, since $A_{k,k}(z)^{-1}$ is meromorphic,

(9) \begin{equation}A_{k,k}(z)^{-1}\text{ is holomorphic at }\alpha\text{ for each }k\text{ such that }\gamma_k\text{ is nonbasic.}\end{equation}

Moreover, by Lemma 4,

(10) \begin{equation}A_{k,k}(\alpha)^{-1}\ll0\text{ for each }k\text{ such that }\gamma_k\text{ is nonbasic.}\end{equation}

If $\zeta(A_{k,k}(\alpha))=0$ then condition (vi) allows us to use Proposition 1 to deduce that $\alpha$ is a simple pole of $A_{k,k}(z)^{-1}$ and that $\eta\lim_{z\to\alpha}(z-\alpha)A_{k,k}(z)^{-1}\gg0$ , where $\eta\in\{-1,1\}$ does not depend on k. Thus

(11) \begin{equation}A_{k,k}(z)^{-1}\text{ has a simple pole at }\alpha\text{ for each }k\text{ such that }\gamma_k\text{ is basic,}\end{equation}

and

(12) \begin{equation}\eta\lim_{z\to\alpha}(z-\alpha)A_{k,k}(z)^{-1}\gg0\text{ for each }k\text{ such that }\gamma_k\text{ is basic.}\end{equation}

We now turn to determining the order of any poles at $\alpha$ of the blocks $B_{j,\, k}(z)$ , $j\leq k$ . The following discussion includes the case $j=k$ , though the behavior of the diagonal blocks may already be understood from (9)– (12). Let $\mathcal{I}(\,j,\, k)$ be the collection of all integer tuples $i=(i_1,\dots,i_\ell)$ with $\ell\in\{1,\dots,k-j+1\}$ and $j=i_1 < \cdots < i_\ell=k$ . Lemma 5 establishes that $B_{j,\, k}(z)=\sum_{i\in\mathcal{I}(\,j,\, k)}D_i(z)$ for all z such that A(z) is nonsingular, where

(13) \begin{align}D_i(z)&\, :\!= \,(\!-1)^{\ell-1}A_{i_1,i_1}(z)^{-1}A_{i_1,i_2}(z)A_{i_2,i_2}(z)^{-1}\cdots A_{i_{\ell-1},i_\ell}(z)A_{i_\ell,i_\ell}(z)^{-1}.\end{align}

For each $i\in\mathcal{I}(\,j,\, k)$ , if $(\gamma_{i_1},\dots,\gamma_{i_\ell})$ is not a chain of classes of $A(\alpha)$ then $\gamma_{i_h}\npreceq\gamma_{i_{h+1}}$ for some h. Under condition (v), we therefore have $A_{i_h,i_{h+1}}(z)=0$ for all $z\in\Omega$ . Consequently,

(14) \begin{equation}D_i(z)=0\text{ if }(\gamma_{i_1},\gamma_{i_2},\dots,\gamma_{i_\ell})\text{ is not a chain.}\end{equation}

If $(\gamma_{i_1},\dots,\gamma_{i_\ell})$ is a length zero chain of classes of $A(\alpha)$ then none of the classes $\gamma_{i_1},\dots,\gamma_{i_\ell}$ are basic. We therefore deduce from (9) that $A_{i_h,i_h}(z)^{-1}$ is holomorphic at $\alpha$ for each h. Consequently,

(15) \begin{equation}D_i(z)\text{ is holomorphic at }\alpha\text{ if }(\gamma_{i_1},\dots,\gamma_{i_\ell})\text{ is a chain of length zero}.\end{equation}

Since $B_{j,\, k}(z)=\sum_{i\in\mathcal{I}(\,j,\, k)}D_i(z)$ for all z such that A(z) is nonsingular, we deduce from (14) and (15) that

(16) \begin{equation} B_{j,\, k}(z) \,\text{has a removable singularity at} \ \alpha \ \text{if no chain from}\ \gamma_j \ \text{to} \ \gamma_k \ \text{has positive length}.\end{equation}

It remains to consider the blocks $B_{j,\, k}(z)$ for which there is a positive length chain from $\gamma_j$ to $\gamma_k$ . We therefore suppose that $i\in\mathcal{I}(\,j,\, k)$ is such that $(\gamma_{i_1},\dots,\gamma_{i_\ell})$ is a chain from $\gamma_j$ to $\gamma_k$ of length $d\geq1$ . In view of (9) and (11) we may define

\begin{align*}F_k\, :\!= \,\begin{cases}\eta\lim_{z\to\alpha}(z-\alpha)A_{k,k}(z)^{-1}&\text{if $\gamma_k$ is basic}\\[3pt]-A_{k,k}(\alpha)^{-1}&\text{if $\gamma_k$ is nonbasic.}\end{cases}\end{align*}

Among $\gamma_{i_1},\dots,\gamma_{i_\ell}$ there are exactly d basic classes and $\ell-d$ nonbasic classes. Therefore, if we multiply both sides of (13) by $({-}1)^{\ell-d}\eta^d(z-\alpha)^d$ and take the limit as $z\to\alpha$ , then we obtain

(17) \begin{align}({-}1)^{\ell-d}\eta^d\lim_{z\to\alpha}(z-\alpha)^d D_i(z)&=({-}1)^{\ell-1}F_{i_1}A_{i_1,i_2}(\alpha)F_{i_2}\cdots A_{i_{\ell-1},i_\ell}(\alpha)F_{i_\ell}.\end{align}

In general the right-hand side of (17) may be zero because if $(\gamma_{i_1},,\dots,\gamma_{i_\ell})$ is not a direct chain then $A_{i_h,i_{h+1}}(\alpha)=0$ for some h. We may nevertheless deduce from (17) that

(18) \begin{align} & D_{i}(z) \text{ is holomorphic at}\ \alpha \ \text{or has a pole of order less than } \textit{d} \ \text{at } \ \alpha \ \text{if} \nonumber \\ &(\gamma_{i_1},\dots,\gamma_{i_\ell}) \ \text{is a chain of length }\,d\, \geq1 \ \text{but not a direct chain.} \end{align}

If $(\gamma_{i_1},\dots,\gamma_{i_\ell})$ is a direct chain then $A_{i_h,i_{h+1}}$ is semipositive for each h. From (12) and (10) we obtain $F_k\gg0$ for each k. Thus, for each h, every column of $A_{i_h,i_{h+1}}(\alpha)F_{i_{h+1}}$ is semipositive. The product of two square matrices with semipositive columns has semipositive columns, so $A_{i_1,i_2}(\alpha)F_{i_2}\cdots A_{i_{\ell-1},i_\ell}(\alpha)F_{i_\ell}$ has semipositive columns. Since $F_{i_1}\gg0$ , we obtain

\begin{equation*}F_{i_1}A_{i_1,i_2}(\alpha)F_{i_2}\cdots A_{i_{\ell-1},i_\ell}(\alpha)F_{i_\ell}\gg0.\end{equation*}

Thus we deduce from (17) that

(19) \begin{align} & D_i(z) \text{ has a pole of order}\ \textit{d} \ \text{at} \ \alpha \ \text{if } (\gamma_{i_1},\dots,\gamma_{i_\ell}) \ \text{ is a direct chain of length } \nonumber \\ &d\geq1. \ \text{ Moreover, }\ \eta^d\lim_{z\to\alpha}(\alpha-z)^dD_i(z)\ll0. \end{align}

Let $d_{j,\, k}$ be the length of the longest chain of classes from $\gamma_j$ to $\gamma_k$ , or let $d_{j,\, k}=0$ if no such chain exists. We noted at the end of Section 3.1 that every chain is nested within a direct chain. Thus if $d_{j,\, k}\geq1$ then $d_{j,\, k}$ is equal to the length of the longest direct chain of classes from $\gamma_j$ to $\gamma_k$ . Let $\mathcal {J}(\,j,\, k)$ be the collection of all integer tuples $i=(i_1,\dots,i_\ell)\in\mathcal{I}(\,j,\, k)$ such that $(\gamma_{i_1},\dots,\gamma_{i_\ell})$ is a direct chain of length $d_{j,\, k}$ . Since $B_{j,\, k}(z)=\sum_{i\in\mathcal{I}(\,j,\, k)}D_i(z)$ , we deduce from (14), (15), (18), and (19) that if $d_{j,\, k}\geq1$ then

\begin{equation*}\eta^{d_{j,\, k}}\lim_{z\to\alpha}(\alpha-z)^{d_{j,\, k}}B_{j,\, k}(z)=\sum_{i\in\mathcal {J}(\,j,\, k)}\eta^{d_{j,\, k}}\lim_{z\to\alpha}(\alpha-z)^{d_{j,\, k}}D_i(z)\ll0.\end{equation*}

Consequently, we find that

(20) \begin{align} & B_{j,\, k}(z) \ \text{has a pole at} \ \alpha \ \text{if there is a positive length chain from} \ \gamma_{j} \ \text{to} \ \gamma_{k}, \ \text{and} \nonumber \\& \text{the order}\ d_{j,\, k} \ \text{of the pole is the length of the longest such chain. Moreover,} \nonumber \\[-2pt]& \eta^{d_{j,\, k}}\lim\nolimits_{z\to\alpha}(\alpha-z)^{d_{j,\, k}}B_{j,\, k}\ll0. \end{align}

The order of the pole of $A(z)^{-1}$ at $\alpha$ is equal to the maximum value attained by $d_{j,\, k}$ over all j,k with $j\leq k$ . By combining (16) and (20), we find that this maximum is equal to the length of the longest chain of classes of $A(\alpha)$ . This proves claim (a).

It remains to prove claims (b) and (c). With a slight abuse of notation, we write $v=[v_1^\top,\dots,v_M^\top]^\top$ and $w=[w_1^\top,\dots,w_M^\top]^\top$ conformably with the $M\times M$ upper triangular block matrix form of $A(z)^{-1}$ . We then have

\begin{align*}v^\top A(z)^{-1}w&=\sum_{j=1}^M\sum_{k=j}^Mv_j^\top B_{j,\, k}(z)w_k.\end{align*}

For each j,k with $j\leq k$ , the function $v_j^\top B_{j,\, k}(z)w_k$ is zero if $v_j=0$ or $w_k=0$ . Suppose instead that $v_j,w_k>0$ . If there is no positive length chain of classes from $\gamma_j$ to $\gamma_k$ , then we deduce from (16) that $\alpha$ is a removable singularity of $v_j^\top B_{j,\, k}(z)w_k$ . If there is a positive length chain of classes from $\gamma_j$ to $\gamma_k$ , then we deduce from (20) that $\alpha$ is a pole of $v_j^\top B_{j,\, k}(z)w_k$ with order $d_{j,\, k}$ equal to the length of the longest such chain, and

\begin{align*}\eta^{d_{j,\, k}}\lim_{z\to\alpha}(\alpha-z)^{d_{j,\, k}}v_j^\top B_{j,\, k}(z)w_k&<0.\end{align*}

Consequently, $\alpha$ is a pole of $v^\top A(z)^{-1}w$ with order equal to the length of the longest chain $(\gamma_{i_1},\dots,\gamma_{i_\ell})$ such that $v_{i_1},w_{i_\ell}>0$ , provided that there exists such a chain with positive length; otherwise, $\alpha$ is a removable singularity of $v^\top A(z)^{-1}w$ . This proves (b) and (c).

To illustrate the role played by conditions (v) and (vi) in Theorem 2, focusing on the key assertion (a), consider the following example with $\boldsymbol{\Omega}=\mathbb C$ , $N=3$ , $\alpha=0$ , and

\begin{equation*}A(z)=\begin{bmatrix}z &\quad 1 & \quad z^2\\[2pt]0 & \quad z &\quad 1\\[2pt]0 & \quad 0 & \quad z \end{bmatrix}\!.\end{equation*}

The matrix A(s) is Metzler for all real s, and the sole eigenvalue of A(0) is 0, so conditions (i) and (ii) of Proposition 1 are satisfied. The directed graph for A(0) is simply $1\to2\to3$ , and thus the classes of A(0) are $\{1\}$ , $\{2\}$ , and $\{3\}$ . All three classes are basic because $A_{1,1}(0)=A_{2,2}(0)=A_{3,3}(0)=0$ . Thus the length of the longest chain of classes of A(0) is 3. Theorem 2 asserts that $A(z)^{-1}$ has a pole of order 3 at 0 if conditions (v) and (vi) are satisfied. Condition (v) is satisfied because each entry of A(z) below the diagonal is identically equal to 0. Condition (vi) is satisfied because each diagonal entry of A ^′(z) is positive. Direct calculation shows that

\begin{equation*}A(z)^{-1}=\dfrac{1}{z^3}\begin{bmatrix}z^2 & \quad -z & \quad 1-z^3\\[2pt]0 & \quad z^2 & \quad -z\\[2pt]0 & \quad 0 & \quad z^2\end{bmatrix}\!,\end{equation*}

confirming that $A(z)^{-1}$ has a pole of order 3 at 0.

Suppose we modify the bottom-left entry of A(z) so that

\begin{equation*}A(z)=\begin{bmatrix}z & \quad 1 & \quad z^2\\[2pt]0 & \quad z & \quad 1\\[2pt] z^2 & \quad 0 & \quad z\end{bmatrix}\!.\end{equation*}

This modification of A(z) satisfies conditions (i) and (i) of Proposition 1 and leaves A(0) unaffected. It satisfies condition (vi) of Theorem 2 but violates condition (v). The violation occurs because the bottom-left entry of A(z) is nonzero for $z\neq0$ , generating a connection from 3 to 1 that unifies the three classes of A(0) into the single class $\{1,2,3\}$ . Direct calculation shows that

\begin{equation*}A(z)^{-1}=\dfrac{-1}{z^2(z^3-z-1)}\begin{bmatrix}z^2 & \quad -z & \quad 1-z^3\\[2pt] z^2 & \quad z^2-z^4 & \quad z\\[2pt] -z^3 & \quad z^2 & \quad z^2\end{bmatrix}\!.\end{equation*}

Thus $A(z)^{-1}$ has a pole of order 2 at 0, while the length of the longest chain of classes of A(0) is 3. This shows that Theorem 2 is not valid if condition (v) is dropped.

Next suppose that instead of modifying the bottom-left entry of A(z) we modify the top-left entry so that

\begin{equation*}A(z)=\begin{bmatrix}z^2 & \quad 1 & \quad z^2\\[2pt]0 & \quad z & \quad 1\\[2pt]0 & \quad 0 & \quad z\end{bmatrix}\!.\end{equation*}

As with the first modification, conditions (i) and (ii) of Proposition 1 continue to be satisfied and A(0) is unaffected. The second modification satisfies condition (v) of Theorem 2 but violates condition (vi). The violation occurs because $A'_{1,1}(0)=0$ . Direct calculation shows that

\begin{equation*}A(z)^{-1}=\dfrac{1}{z^4}\begin{bmatrix}z^2 & \quad -z & \quad 1-z^3\\[2pt] 0 & \quad z^3 & \quad -z^2\\[2pt] 0 & \quad 0 & \quad z^3\end{bmatrix}\!.\end{equation*}

Thus $A(z)^{-1}$ has a pole of order 4 at 0, while the length of the longest chain of classes of A(0) is 3. This shows that Theorem 2 is not valid if condition (vi) is dropped.

The last example leaves open the question of whether condition (vi) can be weakened to require only that $\zeta(A_{\gamma,\gamma}(s))$ has a nonzero left- or right-derivative at $\alpha$ as a function of real $s\in\Omega$ for each basic class $\gamma$ of $A(\alpha)$ . Note that the condition requires the one-sided derivatives to all have the same sign. To illustrate why the same-sign requirement cannot be dropped, consider the map $A\,\colon\, \mathbb C\to\mathbb C^{4\times4}$ defined by

\begin{equation*}A(z)=\begin{bmatrix}z & \quad 1 & \quad 1 & \quad 0\\[2pt]0 & \quad z & \quad 0 & \quad 1\\[2pt]0 & \quad 0 & \quad -z & \quad 1\\[2pt]0 & \quad 0 & \quad 0 & \quad z\end{bmatrix}\!.\end{equation*}

This choice of A(z) satisfies conditions (i) and (ii) of Proposition 1 with $\alpha=0$ , and satisfies condition (v) of Theorem 2. The classes of A(0) are $\{1\}$ , $\{2\}$ , $\{3\}$ , and $\{4\}$ , all of which are basic. The length of the longest chain of classes is 3, achieved by the state transitions $1\to2\to4$ and $1\to3\to4$ . Direct calculation shows that

\begin{equation*}A(z)^{-1}=\dfrac{1}{z^2}\begin{bmatrix}z & \quad -1 & \quad 1 & \quad 0\\[2pt] 0 & \quad z & \quad 0 & \quad -1\\[2pt] 0 & \quad 0 & \quad -z & \quad 1\\[2pt] 0 & \quad 0 & \quad 0 & \quad z\end{bmatrix}\!.\end{equation*}

Thus $A(z)^{-1}$ has a pole of order 2 at 0, not a pole of order 3. The discrepancy arises due to the differing signs of the diagonal entries of A ^′(0), which violates condition (vi).

3.3. The Rothblum index theorem

Theorem 2 bears a striking resemblance to the Rothblum index theorem, a central result in combinatorial spectral theory established in [Reference Rothblum32]. We now elaborate upon the connection between the two results.

Let B be a complex square matrix and $\lambda$ an eigenvalue of B. The index of $\lambda$ is defined to be the smallest nonnegative integer d such that the null space of $(B-\lambda I)^d$ is equal to the null space of $(B-\lambda I)^{d+1}$ . It is necessarily a positive integer no greater than the number of rows or columns of B. As is well known, $\lambda$ is a pole of $(B-z I)^{-1}$ with order equal to the index of $\lambda$ ; see, for instance, Theorem 3.1 and Remark 3.2 in [Reference Campbell and Daners10]. We may therefore equivalently define the index of $\lambda$ to be the order of $\lambda$ as a pole of $(B-zI)^{-1}$ .

The following result, known as the Rothblum index theorem, is part 2 of Theorem 3.1 in [Reference Rothblum32]. It is stated there for a nonnegative matrix B but the extension to Metzler B is immediate.

We now show that Theorem 3 may be obtained as a corollary to Theorem 2 and the Perron–Frobenius theorem.

Proof of Theorem 3. Lemma 1, i.e. the Perron–Frobenius theorem, establishes that $\zeta(B)$ is an eigenvalue of B. To determine the index of this eigenvalue we apply Theorem 2 with $\Omega=\mathbb C$ and $A(z)=B-zI$ . Note that A(z) is somewhere nonsingular because B has finitely many eigenvalues. Also note that adding a real multiple of the identity to a Metzler matrix does not affect the classes of the matrix, the chains of classes of the matrix, or the classification of classes as basic or nonbasic. Thus, for all real $s\in\Omega$ , A(s) and B have the same basic and nonbasic classes and the same chains of classes.

To apply Theorem 2 we need to verify conditions (i) and (ii) of Proposition 1 and conditions (v) and (vi) of Theorem 2. Conditions (i) and (v) are plainly satisfied. Condition (ii) is satisfied for $\alpha=\zeta(B)$ . Condition (vi) is satisfied because if $\gamma$ is a basic class of $A(\alpha)$ , and thus of B, then for every real $s\in\Omega$ with $s\neq\alpha$ we have

\begin{equation*}\dfrac{\zeta(A_{\gamma,\gamma}(s))-\zeta(A_{\gamma,\gamma}(\alpha))}{s-\alpha}=\dfrac{\zeta(B_{\gamma,\gamma})-s-\zeta(A(\alpha))}{s-\alpha}=\dfrac{\zeta(B)-s-0}{s-\alpha}=-1.\end{equation*}

Thus all assumptions of Theorem 2 are satisfied. Conclusion (a) of Theorem 2 asserts that $\alpha$ is a pole of $A(z)^{-1}$ with order d equal to the length of the longest chain of classes of $A(\alpha)$ . That is, $\zeta(B)$ is a pole of $(B-zI)^{-1}$ with order d, meaning that $\zeta(B)$ has index d as an eigenvalue of B. Since $A(\alpha)$ and B have the same basic and nonbasic classes and the same chains of classes, d is equal to the length of the longest chain of classes of B.

The foregoing argument reveals that, insofar as it extends the Perron–Frobenius theorem, the Rothblum index theorem is essentially a special case of Theorem 2 which arises when A(z) is of the affine form $B-zI$ . Keldysh’s theorem for simple eigenvalues provides the bridge which has allowed us to extend the Rothblum index theorem to a more general setting with A(z) holomorphic.

4.1. Stopped Markov additive process

The first model of random growth we consider is the one studied in [Reference Beare, Seo and Toda5]. Time is indexed by $t\in[0,\infty)$ . We are concerned with the growth over time of some real-valued stochastic process $W\, :\!= \,(W_t)_{t\geq 0}$ , say the log-wealth of an economic agent. Growth is random and depends on another stochastic process $J\, :\!= \,(J_t)_{t\geq0}$ taking values in a finite set. The process J may represent, for instance, the time-varying productivity of an economic agent.

We will assume that the bivariate stochastic process (W, J) is a Markov additive process. By this, we mean that the following conditions are satisfied.

1. (i) $\mathbb{P}(W_0=0)=1$ .

2. (ii) W and J have càdlàg paths with probability one.

3. (iii) J is a Markov process with state space $\{1,\ldots,N\}$ for some $N\in\mathbb N$ .

4. (iv) (W, J) is a Markov process with state space $\mathbb R\times\{1,\ldots,N\}$ .

5. (v) $\mathbb{E}[\,f(W_{t+s}-W_t)g(J_{t+s})\mid(W_r,J_r)_{0\leq r\leq t}]=\mathbb{E}_{J_t}[\,f(W_s)g(J_s)]$ for each $s,t>0$ , each Borel measurable function $f\,\colon\, \mathbb R\to\mathbb R$ , and each function $g\,\colon\, \{1,\ldots,N\}\to\mathbb R$ . Here $\mathbb{E}_n$ is the expected value conditional on $J_0=n$ .

This definition of a continuous-time Markov additive process is the same as in [Reference Asmussen1, p. 309] except that we have further required that W is initialized at zero, that paths are càdlàg, and that J has finite state space. If $N=1$ then requiring that (W, J) be a Markov additive process is equivalent to requiring that W be a Lévy process. More generally, W resembles a Lévy process subject to Markov switching. While J remains in state n, W evolves as though it were a Lévy process with Lévy exponent $\varphi_n$ . As discussed in [Reference Asmussen1, pp. 309--310], state transitions trigger a random jump (possibly of negative or zero size) in W. We denote the Laplace transform for the jump when transitioning from state m to state n by $\psi_{m,n}$ . The Lévy exponents $\varphi_n$ and Laplace transforms $\psi_{m,n}$ governing the behavior of W conditional on J, together with the initial state probabilities and infinitesimal generator matrix governing the behavior of J, together determine the joint law of (W, J). See [Reference Asmussen1] and [Reference Beare, Seo and Toda5] for further discussion. We call the process J a Markov modulator.

Each Lévy exponent $\varphi_n$ and Laplace transform $\psi_{m,n}$ is defined on a subset of $\mathbb C$ called its domain. As in Section 1, we define the domain of each Laplace transform $\psi_{m,n}$ to be the set of all $z\in\mathbb C$ such that $\mathbb{E}({\mathrm{e}}^{\operatorname{Re}(z)X})<\infty$ , where X is the random jump in W triggered by a transition from state m to state n. Similarly, we define the domain of each Lévy exponent $\varphi_n$ to be the set of all $z\in\mathbb C$ such that $\mathbb{E}({\mathrm{e}}^{\operatorname{Re}(z)L_1})<\infty$ , where $L=(L_t)_{t\geq0}$ is a Lévy process with Lévy exponent  $\varphi_n$ . The statements about tail probabilities made in the main result of this section, Theorem 4, are vacuous if the intersection of the domains of all $\varphi_n$ and $\psi_{m,n}$ is the imaginary line. Thus we effectively require that either all the Lévy exponents $\varphi_n$ and Laplace transforms $\psi_{m,n}$ correspond to probability distributions with a light upper tail, or all correspond to probability distributions with a light lower tail. We return to this point later.

In economic applications the Lévy exponents $\varphi_n$ and Laplace transforms $\psi_{m,n}$ may be determined by lower-level model parameters. For instance, in the very simple economic model considered in [Reference Beare, Seo and Toda5], $\varphi_n$ and $\psi_{m,n}$ are determined by primitive conditions placed upon the preferences of agents and by an equilibrium condition ensuring that bond markets clear. The manner in which $\varphi_n$ and $\psi_{m,n}$ are determined is immaterial to the present discussion.

The tail probabilities we seek to characterize are those of the random variable $W_T$ , where T is a random time. To explain how T is chosen we need to introduce a subclass of Markov additive processes. Again following [Reference Asmussen1], we say that a Markov additive process $(V,J)\, :\!= \,(V_t,J_t)_{t\geq0}$ is a Markov modulated Poisson process if it satisfies the following conditions.

1. (i) For each state n there exists $\lambda_n\geq0$ such that, for each $s,t>0$ , the distribution of $W_{s+t}-W_s$ conditional on having $J_r=n$ for all $r\in(s,s+t]$ is Poisson with expected value $\lambda_nt$ .

2. (ii) With probability one, V is continuous at each point in time at which J transitions between states.

Thus a Markov modulated Poisson process is simply a Markov additive process for which each state-dependent Lévy exponent is that of a homogeneous Poisson process, and for which state transitions do not trigger jumps. The parameters $\lambda_n$ are referred to as Poisson intensities. We permit Poisson intensities of zero, but a condition to be introduced shortly will require the Poisson intensity to be positive in certain states.

To obtain the stopped process $W_T$ we suppose that in addition to our Markov additive process (W, J) we have a Markov modulated Poisson process (V, J), with W and V sharing the same Markov modulator J, and with W and V conditionally independent given J. We define $T\, :\!= \,\inf\{t\geq0\,\colon\, V_t>0\}$ . Thus the effect is to stop W at a heterogeneous Poisson rate depending on the latent Markov state.

The economic rationale for studying the tail probabilities of $W_T$ can be understood from the following heuristic observation made in [Reference Reed31] for the case $N=1$ : If the wealth of each of a continuum of independent agents in an economy has evolved according to the law of W since the time they were born, and if the distribution of the ages of agents at the present moment is that of T, then the present cross-sectional distribution of wealth is equal to the distribution of $W_T$ . The tail probabilities of $W_T$ may therefore be understood to describe the upper and lower extremes of wealth inequality.

Assumption 1 summarizes the model just described and introduces further notation and technical conditions. Within it we itemize assumptions with Roman numerals and notation with English letters.

We use the following notation to parametrize the law of (W, V, J).

1. (a) $\{1,\dots,N\}$ is the finite state space for J, where $N\in\mathbb N$ .

2. (b) $\Pi\, :\!= \,(\pi_{m,n})$ is the $N\times N$ infinitesimal generator matrix for J.

3. (c) $\varpi\, :\!= \,(\varpi_{n})$ is the $N\times 1$ vector of initial state probabilities for $J_0$ .

4. (d) $\varphi_n$ is the Lévy exponent for W in state n.

5. (e) $\psi_{m,n}$ is the Laplace transform for the jump in W upon transitioning from state m to state n. (If $\pi_{m,n}=0$ then we define $\psi_{m,n}(z)=1$ for all $z\in\mathbb C$ .)

6. (f) $\lambda\, :\!= \,(\lambda_n)$ is the $N\times1$ vector of state-dependent Poisson intensities for V.

We further require that the following conditions are satisfied.

1. (iv) Each initial class of $\Pi$ contains a state n such that $\varpi_n>0$ .

2. (v) Each final class of $\Pi$ contains a state n such that $\lambda_n>0$ .

We further introduce the following notation.

1. (a) $\Omega$ is the intersection of the domains of all $\varphi_n$ and $\psi_{m,n}$ .

2. (b) $\Phi\,\colon\, \Omega\to\mathbb C^{N\times N}$ is the $N\times N$ diagonal matrix-valued function whose nth diagonal entry is the restriction of $\varphi_n$ to $\Omega$ .

3. (c) $\Psi\,\colon\, \Omega\to\mathbb C^{N\times N}$ is the $N\times N$ matrix-valued function whose (m, n)th entry is, for $m\neq n$ , the restriction of $\psi_{m,n}$ to $\Omega$ . Each diagonal entry of $\Psi$ is one.

4. (d) $\Lambda$ is the $N\times N$ diagonal matrix with nth diagonal entry equal to $\lambda_n$ .

5. (e) $A\,\colon\, \Omega\to\mathbb C^{N\times N}$ is the $N\times N$ matrix-valued function defined by

   \begin{equation*}A(z)=\Phi(z)+\Pi\odot\Psi(z)-\Lambda,\end{equation*}
   where $\odot$ is the Hadamard (entry-wise) product.

We say that a subset of $\mathbb C$ is a strip in the complex plane if it is equal to $\{z\in\mathbb C\,\colon\! \operatorname{Re}(z)\in C\}$ for some nonempty convex set $C\subseteq\mathbb R$ . The domain of every Laplace transform, and thus of every Lévy exponent, is a strip in the complex plane containing zero. Thus the set $\Omega$ defined in part (a) of Assumption 1 is a strip in the complex plane containing zero.

The only conditions imposed in Assumption 1 not already discussed are parts (iv) and (v). These conditions do not appear explicitly in [Reference Beare, Seo and Toda5], but are implicitly present. Condition (iv) amounts to excluding redundant Markov states; that is, states which are never visited with probability one. The same is done in [Reference Beare, Seo and Toda5, p. 990]. Condition (v) is used to guarantee that $\mathbb{P}(T<\infty)=1$ , which is assumed directly at several points in [Reference Beare, Seo and Toda5]. Condition (v) also ensures that $\zeta(A(0))<0$ , which will be important later.

Our main result in this section is the following characterization of the tail probabilities of $W_T$ . As in Section 1, we denote by $\Omega^\circ$ the interior of $\Omega$ .

Theorem 4. Let $(W_t,V_t,J_t)_{t\geq0}$ be a stochastic process satisfying Assumption 1. Define $T\, :\!= \,\inf\{t\geq0\,\colon\, V_t>0\}$ , noting that $\mathbb{P}(T<\infty)=1$ by Lemma 6. If there exists a positive real number $\alpha$ in $\Omega^\circ$ such that $\zeta(A(\alpha))=0$ , then

(21) \begin{equation}0<\liminf_{w\to\infty}w^{-d_\alpha+1}{\mathrm{e}}^{\alpha w}\mathbb{P}(W_T>w)\leq\limsup_{w\to\infty}w^{-d_\alpha+1}{\mathrm{e}}^{\alpha w}\mathbb{P}(W_T>w)<\infty,\end{equation}

where $d_\alpha$ is the length of the longest chain of classes of $A(\alpha)$ . If there exists a negative real number $-\beta$ in $\Omega^\circ$ such that $\zeta(A(\!-\!\beta))=0$ , then

(22) \begin{equation}0<\liminf_{w\to\infty}w^{-d_{-\beta}+1}{\mathrm{e}}^{\beta w}\mathbb{P}(W_T<-w)\leq\limsup_{w\to\infty}w^{-d_{-\beta}+1}{\mathrm{e}}^{\beta w}\mathbb{P}(W_T<-w)<\infty,\end{equation}

where $d_{-\beta}$ is the length of the longest chain of classes of $A(\!-\!\beta)$ .

Theorem 4 is a generalization of Theorem 3.1 in [Reference Beare, Seo and Toda5]. The latter result requires $\Pi$ to be irreducible. In this case $A(\alpha)$ and $A(\!-\!\beta)$ are also irreducible, so that $d_\alpha=d_{-\beta}=1$ , and the multiplicative factors $w^{-d_\alpha+1}$ and $w^{-d_{-\beta}+1}$ vanish. Thus the tail probabilities of $W_T$ resemble those of an exponential distribution. When $\Pi$ is reducible this need not be the case. Recalling our discussion of exponential versus Erlang tail probabilities in Section 1, we learn from Theorem 4 that the tail probabilities of $W_T$ in general resemble those of an Erlang distribution, and learn how the parameters of the relevant Erlang distribution (i.e. $d_\alpha$ and $\alpha$ for the upper tail, and $d_{-\beta}$ and $\beta$ for the lower tail) are determined by the parameters of the random growth model.

The Erlang parameters $\alpha$ and $\beta$ for the upper and lower tail probabilities are determined by the equation $\zeta(A(s))=0$ . The characterization of upper tail probabilities provided in Theorem 4 requires this equation to admit a positive solution in $\Omega^\circ$ , while the characterization of lower tail probabilities requires it to admit a negative solution in $\Omega^\circ$ . Of course, a necessary condition to have a positive or negative solution is that there exists some positive or negative number in  $\Omega^\circ$ . Thus, as alluded to earlier, our characterization of upper (lower) tail probabilities requires all of the Lévy exponents $\varphi_n$ and Laplace transforms $\psi_{m,n}$ to include positive (negative) real numbers in their domain, and thus be associated with probability distributions which have a light upper (lower) tail. This is natural: an Erlang distribution is itself light-tailed, so we cannot expect $W_T$ to have an upper or lower tail resembling that of an Erlang distribution if the corresponding tail of $W_t$ is not light for each fixed t.

If either a positive or negative solution to the equation $\zeta(A(s))=0$ exists in $\Omega^\circ$ , then it is the unique positive or negative solution in $\Omega^\circ$ . This is a consequence of the fact that $\zeta(A(0))<0$ (by Lemma 6) and the fact that $\zeta(A(s))$ is convex as a function of real $s\in\Omega$ . The latter fact is an implication of the following result of Nussbaum [Reference Nussbaum27]; see inequalities (1.2) and (1.4) in Theorem 1.1 therein.

Convexity of $\zeta(A(s))$ follows from Lemma 7 by taking $F(s)=\Pi\odot\Psi(s)$ and $G(s)=\Phi(s)-\Lambda$ . Note that every Laplace transform is log-convex on the real part of its domain, and every Lévy exponent is convex on the real part of its domain. The fact that $\zeta(A(s))$ is a convex function of s depending in a simple way on the parameters of our random growth model makes it easy to determine in specific applications whether $\alpha$ and/or $\beta$ exist. We refer to [Reference Beare, Seo and Toda5] for further discussion of the existence of $\alpha$ and $\beta$ , including a simple sufficient condition for existence.

Our proof of Theorem 4 involves verifying that the Laplace transform for $W_T$ behaves at $\alpha$ and/or $-\beta$ in a manner permitting the application of Theorem 1. Proposition 2.2 in [Reference Asmussen1, p. 311] establishes that the Laplace transform for $W_t$ , with t fixed, satisfies

\begin{equation*}\mathbb{E}({\mathrm{e}}^{zW_t})=\varpi^\top\exp(t(\Phi(z)+\Pi\odot\Psi(z)))1_N\end{equation*}

on its domain, where $\exp$ is the matrix exponential function and $1_N$ is an $N\times1$ vector of ones. Building on this result, Propositions 3.3 and 3.4 in [Reference Beare, Seo and Toda5] establish (without assuming irreducibility of $\Pi$ ) the following characterization of the Laplace transform for $W_T$ . We do not repeat the proof here.

Lemma 8. Let W and T be as in Theorem 4. Let z be a point in $\Omega$ such that $\zeta(A(\operatorname{Re}(z)))<0$ . Then A(z) is nonsingular, and

\begin{equation*}\mathbb{E}({\mathrm{e}}^{zW_T})=-\varpi^\top A(z)^{-1}\lambda.\end{equation*}

Note that the formula provided in Proposition 3.4 in [Reference Beare, Seo and Toda5] is $\mathbb{E}({\mathrm{e}}^{zW_T})=\varpi^\top A(z)^{-1}A(0)1_N$ . This is equivalent to the formula provided in Lemma 8 because $A(0)=\Pi-\Lambda$ , $\Pi1_N=0$ , and $\Lambda1_N=\lambda$ .

We require one further lemma, easily proved, for our proof of Theorem 4.

The second assertion of Lemma 9 implies that the classes and chains of classes of A(s) are the same for all real $s\in\Omega$ . Note, however, that the classification of classes as basic or nonbasic, and therefore the length of chains, may vary with s.

Proof of Theorem 4. Suppose there exists a positive real number $\alpha$ in $\Omega^\circ$ such that $\zeta(A(\alpha))=0$ . We must show that in this case (21) is satisfied. Since $\Omega$ is a strip in the complex plane containing 0 and $\alpha$ , it must be the case that $\Omega^\circ$ is a strip in the complex plane satisfying $\Omega_{0,\alpha}\subseteq\Omega^\circ$ , where we define $\Omega_{0,\alpha}\, :\!= \,\{z\in\mathbb C\,\colon\, 0<\operatorname{Re}(z)<\alpha\}$ . The function A(z) is holomorphic on $\Omega^\circ$ because Laplace transforms, and thus Lévy exponents, are holomorphic on the interior of their domains. If A(z) is also somewhere nonsingular on $\Omega^\circ$ then, by the first part of Proposition 1, $A(z)^{-1}$ is meromorphic on $\Omega^\circ$ . In fact, we have $\zeta(A(s))<0$ for all $s\in(0,\alpha)$ because $\zeta(A(0))<0$ (by Lemma 6) and because $\zeta(A(s))$ is a convex function of real $s\in\Omega$ (by Lemma 7 and the remark immediately following). Thus A(z) is nonsingular for all $z\in\Omega_{0,\alpha}$ by Lemma 8, and $A(z)^{-1}$ is meromorphic on $\Omega^\circ$ . Consequently, we may define a function f(z) meromorphic on $\Omega^\circ$ by setting

\begin{equation*}f(z)=\varpi^\top A(z)^{-1}\lambda.\end{equation*}

We apply part (b) of Theorem 2 to f(z), with $v=\varpi$ and $w=\lambda$ . If the assumptions of Theorem 2 are satisfied, then it asserts that if there exists a positive length chain of classes $(\gamma_1,\dots,\gamma_\ell)$ of $A(\alpha)$ such that $\max_{m\in\gamma_1}\varpi_m>0$ and $\max_{n\in\gamma_\ell}\lambda_n>0$ then $\alpha$ is a pole of f(z) with order equal to the length of the longest such chain. The initial and final classes of $A(\alpha)$ are the same as those of $\Pi$ by Lemma 9, so condition (iv) in Assumption 1 makes the restriction to chains for which $\max_{m\in\gamma_1}\varpi_m>0$ redundant, while condition (v) in Assumption 1 makes the restriction to chains for which $\max_{n\in\gamma_\ell}\lambda_n>0$ redundant. Thus, if the assumptions of Theorem 2 are satisfied, then $\alpha$ is a pole of f(z) with order equal to $d_\alpha$ , the length of the longest chain of classes of $A(\alpha)$ .

The assumptions we need to verify are conditions (i) and (ii) of Proposition 1 and conditions (v) and (vi) of Theorem 2. The first two conditions are plainly satisfied. Condition (v) of Theorem 2 is satisfied because if $\gamma$ and $\delta$ are classes of $A(\alpha)$ with $\gamma\npreceq\delta$ then $A_{\gamma,\delta}(\alpha)=0$ , and thus $A_{\gamma,\delta}(z)=0$ for every $z\in\Omega$ by the first assertion of Lemma 9. It remains to verify condition (vi) of Theorem 2. Let $\gamma$ be a basic class of $A(\alpha)$ . Since $\gamma$ is basic we have $\zeta(A_{\gamma,\gamma}(\alpha))=\zeta(A(\alpha))=0$ . The classes of $A(\alpha)$ are the same as those of A(0) by the second assertion of Lemma 9. Thus $\gamma$ is a class of A(0), implying that $\zeta(A_{\gamma,\gamma}(0))\leq\zeta(A(0))$ . By Lemma 6 we therefore have $\zeta(A_{\gamma,\gamma}(0))<0$ . Moreover, $\zeta(A_{\gamma,\gamma}(s))$ is convex as a function of real $s\in\Omega$ by Lemma 7, taking $F(s)=\Pi_{\gamma,\gamma}\odot\Psi_{\gamma,\gamma}(s)$ and $G(s)=\Phi_{\gamma,\gamma}(s)-\Lambda_{\gamma,\gamma}$ . Since $\zeta(A_{\gamma,\gamma}(s))$ is convex in s, negative at $s=0$ , and zero at $s=\alpha$ , it must have positive left- and right-derivatives at $s=\alpha$ . Thus condition (vi) of Theorem 2 is satisfied. We conclude that $\alpha$ is a pole of f(z) with order $d_\alpha$ .

Let $\varphi$ denote the Laplace transform for $W_T$ , and let $\Omega_\varphi$ denote its domain, that is,

\begin{equation*}\Omega_\varphi\, :\!= \,\bigl\{z\in\mathbb C\,\colon\, \mathbb{E}\bigl({\mathrm{e}}^{\operatorname{Re}(z)W_T}\bigr)<\infty\bigr\}.\end{equation*}

Let $\varphi^\circ$ denote the restriction of $\varphi$ to $\Omega_\varphi^\circ$ , a holomorphic function. Recalling that $\zeta(A(s))<0$ for all $s\in(0,\alpha)$ , we deduce from Lemma 8 that $\Omega_{0,\alpha}\subseteq\Omega_\varphi^\circ$ and that $\varphi^\circ$ coincides with $-f$ on $\Omega_{0,\alpha}$ . Since f is meromorphic on an open neighborhood of $\alpha$ with a pole of order $d_\alpha$ at $\alpha$ , it follows that $\alpha$ is the right abscissa of convergence of $\varphi$ , and that $\varphi^\circ$ can be meromorphically extended to an open set containing $\alpha$ in such a way that $\alpha$ is a pole of order $d_\alpha$ of this extension. Thus, by applying Theorem 1, we obtain the characterization of upper tail probabilities in (21).

A symmetric argument can be used to establish the characterization of lower tail probabilities in (22) when there exists a negative $-\beta\in\Omega^\circ$ such that $\zeta(A(\!-\!\beta))=0$ .

We close this subsection with a simple example illustrating the application of Theorem 4. Let $(W,J,V)_{t\geq 0}$ be a stochastic process satisfying Assumption 1 with infinitesimal generator matrix $\Pi$ and initial state probability vector $\varpi$ satisfying

\begin{equation*}\Pi=\begin{bmatrix}-\theta_1& \quad\theta_1& \quad0& \quad\cdots& \quad0& \quad0\\[2pt]0& \quad-\theta_2& \quad\theta_2& \quad\cdots& \quad0& \quad0\\[2pt]\vdots& \quad\vdots& \quad\vdots& \quad& \quad\vdots& \quad\vdots\\[2pt]0& \quad0& \quad0& \quad\cdots& \quad-\theta_{N-1}& \quad\theta_{N-1}\\[2pt]0& \quad0& \quad0& \quad\cdots& \quad0& \quad0\end{bmatrix}\quad\text{and}\quad\varpi=\begin{bmatrix}1\\[2pt]0\\[2pt]\vdots\\[2pt]0\\[2pt]0\end{bmatrix}\!,\end{equation*}

where $\theta_1,\dots,\theta_{N-1}$ are strictly positive. The directed graph for $\Pi$ is simply $1\to2\to\cdots\to N$ . The Markov modulator J begins in state 1 with probability $\varpi_1=1$ and transitions from each state to the subsequent state at rate $\pi_{n,n+1}=\theta_n$ . The stopping rates $\lambda_1,\dots,\lambda_{N-1}$ are arbitrary nonnegative numbers and the final stopping rate $\lambda_N$ is strictly positive. Assume that state transitions do not trigger jumps, so that every entry of $\Psi(z)$ is identically equal to one. Let $c_n=\theta_n+\lambda_n$ for $n=1,\dots,N-1$ , and let $c_N=\lambda_N$ . Then

\begin{equation*}A(z)=\begin{bmatrix}\varphi_1(z)-c_1& \quad\theta_1& \quad\cdots& \quad0& \quad0\\[2pt]0& \quad\varphi_2(z)-c_2& \quad\cdots& \quad0& \quad0\\[2pt]\vdots& \quad\vdots& \quad\ddots& \quad\vdots& \quad\vdots\\[2pt]0& \quad0& \quad\cdots& \quad\varphi_{N-1}(z)-c_{N-1}& \quad\theta_{N-1}\\[2pt]0& \quad0& \quad\cdots& \quad0& \quad\varphi_N(z)-c_N\end{bmatrix}\!.\end{equation*}

Assume that each state-dependent Lévy exponent $\varphi_n$ takes the Gaussian form $\varphi_n(z)=\mu_nz+\sigma_n^2z^2/2$ with state-dependent mean $\mu_n$ and state-dependent variance $\sigma^2_n>0$ . The unique positive value of $\alpha_n$ such that $\varphi_n(\alpha_n)=c_n$ is

\begin{equation*}\alpha_n=\dfrac{-\mu_n+\sqrt{\mu_n^2+2\sigma_n^2c_n}}{\sigma_n^2},\end{equation*}

and for real $s\in(0,\alpha_n)$ we have $\varphi_n(s) < c_n$ . Let $\alpha=\min_{n}\alpha_n$ and let $d_\alpha$ be the number of states for which $\alpha_n=\alpha$ . Then $\zeta(A(\alpha))=0$ , and there are exactly $d_\alpha$ zeros along the diagonal of $A(\alpha)$ , with the other diagonal entries of $A(\alpha)$ being negative. The classes and chains of classes for $A(\alpha)$ are the same as those for $\Pi$ , and exactly $d_\alpha$ of the N classes of $A(\alpha)$ are basic. Thus Theorem 4 shows that the upper tail probabilities of the stopped Markov additive process satisfy (21), that is, the upper tail probabilities are Erlang-like with rate parameter $\alpha$ and shape parameter $d_\alpha$ . A similar argument using Theorem 4 shows that the lower tail probabilities of the stopped Markov additive process satisfy (22) with $-\beta$ equal to the maximum value of

\begin{equation*}-\beta_n=\dfrac{-\mu_n-\sqrt{\mu_n^2+2\sigma_n^2c_n}}{\sigma_n^2}\end{equation*}

across states and with $d_{-\beta}$ equal to the number of states for which $\beta_n=\beta$ .

4.2. Discrete-time Markov additive process with reset

Our second model of random growth is the one studied in [Reference Beare and Toda3]. It is in many ways similar to the model treated in the previous subsection, but with two key differences. First, time is discrete, taking values in $\mathbb N_0\, :\!= \,\{0\}\cup\mathbb N$ . Second, rather than study the tail probabilities of a random variable obtained by stopping our growth process at a random time, we endow our growth process with a resetting mechanism inducing a unique stationary distribution, and study the tail probabilities of this stationary distribution. The resetting mechanism is intended to model the random destruction and creation of agents in an economy. This modeling device originates in work by Yaari and Blanchard [Reference Blanchard9, Reference Yaari39] and is commonly employed in models of stationary economies with overlapping generations of agents subject to random mortality. See [Reference Beare and Toda4] for a recent application related to the optimal choice of taxation policy.

Assumption 2 defines our second random growth model and introduces associated notation. As in the statement of Assumption 1, we use Roman numerals to itemize required conditions and English letters to itemize notation.

We further introduce the following notation.

1. (a) $\varphi_{m,n}$ is the Laplace transform for $\varepsilon_1(m,n)$ . (If $\mathbb{P}(J_0=m,J_1=n)=0$ then we may assume without loss of generality that $\varepsilon_1(m,n)=0$ , so that $\varphi_{m,n}(z)=1$ for all $z\in\mathbb C$ .)

2. (b) $\Omega$ is the intersection of the domains of all $\varphi_{m,n}$ .

3. (c) $\Phi\,\colon\, \Omega\to\mathbb C^{N\times N}$ is the $N\times N$ matrix-valued function whose (m, n)-entry is the restriction of $\varphi_{m,n}$ to $\Omega$ .

4. (d) $A\,\colon\, \Omega\to\mathbb C^{N\times N}$ is the $N\times N$ matrix-valued function defined by

   \begin{equation*}A(z)=\Pi\odot\Upsilon\odot\Phi(z)-I,\end{equation*}
   where I is the $N\times N$ identity matrix.

Given $\Pi$ , $\Upsilon$ , and $\varpi$ , parts (i) and (ii) of Assumption 2 determine the Markov transition probabilities for $(V_t,J_t)_{t\in\mathbb N_0}$ . Note that if all entries of $\Upsilon$ were equal to one – which is not permitted under condition (v) – then $(J_t)_{t\in\mathbb N_0}$ would be a stationary Markov chain with transition probability matrix $\Pi$ , and we would have $V_t=0$ for all t with probability one. To the contrary, condition (v) guarantees that each final class of $\Pi$ contains a state m such that $\mathbb{P}(V_{t+1}=1\mid J_t=m)>0$ , thereby ensuring that $V_t=1$ infinitely often with probability one. Part (iii) of Assumption 2 requires $(W_t)_{t\in\mathbb N_0}$ to increase by $\varepsilon_{t+1}(J_{t},J_{t+1})$ from period t to period $t+1$ if $V_{t+1}=0$ , or to reset to zero if $V_{t+1}=1$ . Condition (iv) excludes the presence of states which, with probability one, are never visited by $J_t$ .

If $(W_t,V_t,J_t)_{t\in\mathbb N_0}$ satisfies Assumption 2 then $({\mathrm{e}}^{W_t},V_t,J_t)_{t\in\mathbb N_0}$ is a Markov multiplicative process with reset in the sense defined in [Reference Beare and Toda3]. The characterization of the upper and lower tails of the time-invariant distribution of ${\mathrm{e}}^{W_t}$ established in [Reference Beare and Toda3] – see in particular Theorem 3 therein – may be adapted in an obvious way to apply to the time-invariant distribution of $W_t$ . However, it is required in [Reference Beare and Toda3] that the matrix $\Pi\odot\Upsilon$ be irreducible. Our main result in this section, which may be compared to Theorem 4 in Section 4.1, does not impose this requirement, and consequently provides a more general characterization of tail probabilities.

Theorem 5. Let $(W_t,V_t,J_t)_{t\in\mathbb N_0}$ be a stationary Markov chain satisfying Assumption 2. If there exists a positive real number $\alpha$ in $\Omega^\circ$ such that $\zeta(A(\alpha))=0$ , then

\begin{equation*}0<\liminf_{w\to\infty}w^{-d_\alpha+1}{\mathrm{e}}^{\alpha w}\mathbb{P}(W_0>w)\leq\limsup_{w\to\infty}w^{-d_\alpha+1}{\mathrm{e}}^{\alpha w}\mathbb{P}(W_0>w)<\infty,\end{equation*}

where $d_\alpha$ is the length of the longest chain of classes of $A(\alpha)$ . If there exists a negative real number $-\beta$ in $\Omega^\circ$ such that $\zeta(A(\!-\!\beta))=0$ , then

\begin{equation*}0<\liminf_{w\to\infty}w^{-d_{-\beta}+1}{\mathrm{e}}^{\beta w}\mathbb{P}(W_0<-w)\leq\limsup_{w\to\infty}w^{-d_{-\beta}+1}{\mathrm{e}}^{\beta w}\mathbb{P}(W_0<-w)<\infty,\end{equation*}

where $d_{-\beta}$ is the length of the longest chain of classes of $A(\!-\!\beta)$ .

We will prove Theorem 5 using arguments similar to those we used to prove Theorem 4. The following lemma plays a similar role in the proof of Theorem 5 to that of Lemma 6 in the proof of Theorem 4.

The next lemma, which may be compared to Lemma 8, is an immediate consequence of Lemma 2 in [Reference Beare and Toda3]. We do not repeat the proof here.

Lemma 11. Let $(W_t,V_t,J_t)_{t\in\mathbb N_0}$ be a stationary Markov chain satisfying Assumption 2. Let z be a point in $\Omega$ such that $\zeta(A(\operatorname{Re}(z)))<0$ . Then A(z) is nonsingular, and

\begin{align*}\mathbb{E}({\mathrm{e}}^{zW_0})&=-\mathbb{P}(V_0=1)\varpi^\top A(z)^{-1}1_N.\end{align*}

The final lemma we require for our proof of Theorem 5 resembles Lemma 9 above. We omit the proof as it is an obvious modification to the proof of Lemma 9.

Proof of Theorem 5. The proof is similar to the proof of Theorem 4. We again prove only the characterization of upper tail probabilities. A symmetric argument can be used to prove the characterization of lower tail probabilities.

Suppose there exists a positive real $\alpha\in\Omega^\circ$ such that $\zeta(A(\alpha))=0$ , and define $\Omega_{0,\alpha}\, :\!= \,\{z\in\mathbb C\,\colon\, 0<\operatorname{Re}(z)<\alpha\}$ . Arguments given at the beginning of the proof of Theorem 4 establish that $\Omega_{0,\alpha}\subseteq\Omega^\circ$ , and that $A(z)^{-1}$ is meromorphic on $\Omega^\circ$ if A(z) is nonsingular somewhere on $\Omega^\circ$ . By applying Lemma 7 with $F(s)=\Pi\odot\Upsilon\odot\Phi(s)$ and $G(s)=-I$ , we find that $\zeta(A(s))$ is a convex function of real $s\in\Omega$ . Therefore, since $\zeta(A(0))<0$ by Lemma 10, and $\zeta(A(\alpha))=0$ , we must have $\zeta(A(s))<0$ for all $s\in(0,\alpha)$ . Thus $A(z)^{-1}$ is meromorphic on $\Omega^\circ$ , and we may define a function f(z) meromorphic on $\Omega^\circ$ by setting

\begin{equation*}f(z)=\varpi^\top A(z)^{-1}1_N.\end{equation*}

We apply part (b) of Theorem 2 to f(z), with $v=\varpi$ and $w=1_N$ . If the assumptions of Theorem 2 are satisfied, then it asserts that if there exists a positive length chain of classes $(\gamma_1,\dots,\gamma_\ell)$ of $A(\alpha)$ such that $\max_{m\in\gamma_1}\varpi_m>0$ then $\alpha$ is a pole of f(z) with order equal to the length of the longest such chain. The initial classes of $A(\alpha)$ are the same as those of $\Pi\odot\Upsilon$ by the second assertion of Lemma 12, so condition (iv) in Assumption 2 makes the restriction to chains for which $\max_{m\in\gamma_1}\varpi_m>0$ redundant. Thus, if the assumptions of Theorem 2 are satisfied, then $\alpha$ is a pole of f(z) with order equal to $d_\alpha$ , the length of the longest chain of classes of $A(\alpha)$ . We may verify these assumptions in the same way as in the proof of Theorem 4, but using Lemma 12 in place of Lemma 9, using Lemma 10 in place of Lemma 6, and applying Lemma 7 with $F(s)=\Pi_{\gamma,\gamma}\odot\Upsilon_{\gamma,\gamma}\odot\Phi_{\gamma,\gamma}(s)$ and $G(s)=-I$ .

Let $\varphi$ denote the Laplace transform for $W_0$ , and let $\Omega_\varphi$ denote its domain, that is,

\begin{equation*}\Omega_\varphi\, :\!= \,\{z\in\mathbb C\,\colon\, \mathbb{E}({\mathrm{e}}^{\operatorname{Re}(z)W_0})<\infty\}.\end{equation*}

Let $\varphi^\circ$ denote the restriction of $\varphi$ to $\Omega_\varphi^\circ$ , a holomorphic function. Recalling that $\zeta(A(s))<0$ for all $s\in(0,\alpha)$ , we deduce from Lemma 11 that $\Omega_{0,\alpha}\subseteq\Omega_\varphi^\circ$ and that $\varphi^\circ$ coincides with $-\mathbb{P}(V_0=1)\,f$ on $\Omega_{0,\alpha}$ . We know that $\mathbb{P}(V_0=1)>0$ because $(V_t)_{t\in\mathbb N_0}$ is stationary and condition (v) in Assumption 2 implies that $V_t=1$ infinitely often with probability one. Therefore, since f is meromorphic on an open neighborhood of $\alpha$ with a pole of order $d_\alpha$ at $\alpha$ , it follows that $\alpha$ is the right abscissa of convergence of $\varphi$ , and that $\varphi^\circ$ can be meromorphically extended to an open set containing $\alpha$ in such a way that $\alpha$ is a pole of order $d_\alpha$ of this extension. Thus, by applying Theorem 1, we obtain the asserted characterization of upper tail probabilities.