From the definition of matrix-geometric distributions it is immediately clear that the relation
DPH⊆MG
must hold. The question arises though whether or not DPH (MG ? In the continuous case it was easy to find an example of a distribution that is matrix-exponential but not of phase-type, as phase-type distributions can never be zero on (0,∞). But this is not the case for discrete phase-type distributions, as they can have a periodicity which allows them to be zero with a certain periodd.
In ’Characterization of phase-type distributions’ by C.A. O’Cinneide,[14], the following characterization of discrete phase-type distributions is given.
4.2 Existence of genuine matrix-geometric distributions 35
Theorem 4.3 (Characterization of discrete phase-type distributions) A probability measure µ on {0,1, . . .} with rational generating function is of phase type if and only if for some positive integer ω each of the generating functions
Hµ,k(z) =
∞
X
n=0
µ(ωn+k)zn, (4.3)
for k= 0,1, . . . , ω−1, is either a polynomial or has a unique pole of minimal absolute value.
The proof of this theorem can be found in [14], pages 49-56. Here, we will give an outline of the proof.
Necessity. Let µbe discrete phase-type. Then either µis the point mass at zero, for which Theorem 4.3 holds with ω = 1 or µ has an irreducible repre-sentation (α, T) of positive order. In the latter case, we let {Xn}n∈N be the Markov chain corresponding to this representation. Let ω be the least com-mon multiple of the periods of all the periodic classes of the chain X. Let τ be the time until absorption of the Markov chain, hence τ has distribution µ. If P(τ=kmodω) = 0 then Hµ,k(z) is identically zero and hence a poly-nomial. If however P(τ=kmodω) > 0 we can define a new Markov chain {Yn}={Xk+ωn}which given that τ =kmodω is an absorbing Markov chain with one absorbing state and possibly fewer transient states than our original chain since we can leave out the states than can never be reached.
If we denote by αk the initial distribution ofY0 and by Sk the transition pro-bability matrix of its transient states we have that (αk, Sk) is an irreducible representation of the conditional distribution of τ−kω givenτ =kmodω.By the definition of ω,the transition matrixSk has no periodic states. If we call this distributionµk we have thatHµk(z) = HHµ,k(z)
µ,k(1) for allz.
In order to prove that Hµ,k has a unique pole of minimal value it suffices now to prove thatHµk(z) has one, sinceHµ,k(z) is just a positive multiple ofHµk(z) and their poles are equal. Letζk denote the unique eigenvalue of Sk of largest absolute value, whose existence is assured by the Perron-Frobenius theorem (see e.g. [8], page 240). The uniqueness of ζk is a consequence of Sk being aperi-odic which implies that it cannot have non-real eigenvalues of maximal modulus (see Thm. 3.3.1 in [11]). SinceSk is a transition probability matrix, ζk is less than one. As the poles ofHµk(z) are the reciprocals of the eigenvalues ofSk, it follows ζ1
k is the unique pole of minimal value ofHµk(z) and hence ofHµ,k(z).
The other poles ofHµ,k(z) will have greater absolute value, which concludes the proof of necessity.
Sufficiency. Let µ be a probability measure on {0,1, . . .} with rational gen-erating function and with each of the gengen-erating functions Hµ,k(z) either a polynomial or with a unique pole of minimal absolute value. In [14] it is proven that a probability measure with rational generating function with a unique pole of minimal absolute value is of discrete phase-type. The proof of this statement makes use of invariant polytopes and is beyond the scope of this thesis.
However, this statement is the crux of the proof for the sufficiency of the condi-tion to be of discrete phase-type. We can finish the proof with the following ar-gument. The probability measureµkwith generating functionHµk(z) = HHµ,k(z)
µ,k(1)
is of discrete phase-type asHµk(z) has a unique pole of minimal absolute value.
Letτk be a random variable with distribution µk fork= 0,1, . . . , ω−1.Then ωτk is again of discrete phase-type, as we can modify the state space such that from every original state there is a deterministic series of ω −1 transitions throughω−1 new states after which again an original state is reached. As a constantc can be modelled as a phase-type distribution with transition matrix of dimensioncwithTi,i+1= 1 for all statesiwe have that alsoωτk+kis discrete phase-type distributed. Nowµ is a mixture of the distributionsµk forωτk+k in proportions ofHµ,k(1) fork={0,1, . . . , ω−1}. Hence µis a finite mixture of discrete phase-type distributions and is so itself of discrete phase-type, which
completes the proof for sufficiency.
Based on this characterization of discrete phase-type distributions we can prove the following theorem:
Theorem 4.4 The class of matrix-geometric distributions is strictly larger than the class of discrete phase-type distributions.
Proof. We prove this theorem by giving an example of a matrix-geometric distribution that does not meet the necessary condition stated in Theorem 4.3 to be of discrete phase-type. We have a look at a matrix-geometric distribution with density
g(n) =Kpn−1
1 + cos(√
2(n−1))
, n∈N (4.4)
forK a normalizing constant andp∈(0,1).We can rewrite this density to get it into matrix-geometric form:
g(n) = Kpn−1
1 + cos√
2(n−1)
= Kpn−1
1 + 1 2ei
√2(n−1)+1 2e−i
√2(n−1)
4.2 Existence of genuine matrix-geometric distributions 37
We write as usualg(n) =αTn−1t.According to Theorem 4.3 a necessary condi-tion for this density to be of discrete phase-type is that the probability genera-ting functionsHg,k(z) =P∞ We see that the zeros of the denominator of these generating functions are given by the reciprocals of the eigenvalues of the matrixTω.We have to check for Formula (4.7) that none of these zeros cancel. We can use the representation
ofg(n) in (4.5) to find which are the reciprocals of the eigenvalues of the matrix
Tω=
and which are not a zero of the numerator ofHg,0(z).These poles all have equal absolute value p−ω. The only possibility for Hg,k(z) to have a unique pole of maximum absolute value is for ω=k√
2πbecause thenei
√2ω=ek2πi= 1 and Tω will have eigenvalue pk
√2π with multiplicity three. However, according to Theorem 4.3, ω must be chosen in N, which proves that it is not possible for Hg,k(z) to have a unique pole of minimal absolute value.
Finally, we know thatHg,k(z) is not the zero polynomial since g(n) is positive for alln∈N, which is a consequence of (1 + cos(√
2(n−1)) not having a zero in the integer numbers.
We can conclude that the distribution represented byg(n) is genuinely matrix-geometric and that the class of matrix-matrix-geometric distributions is strictly larger than the class of discrete phase-type distributions.
Example 4.1 The density
g(n) = 1 is genuinely matrix-geometric. Figure 4.1 shows the density plot ofg(n).