Rational arrival processes with strictly positive densities need not be Markovian

Rational arriv al pro cesses with strictly p ositiv e densities need not b e Mark o vian Oscar P eralta Abstract T elek [ 12 ] ask ed whether a rational arriv al process (RAP), sp ecified by matrices G 0 and G 1 and an initial ro w v ector ν , with strictly positive joint densities and a unique dominan t real eigen v alue of G 0 m ust admit an equiv alent Mark ovian arriv al process (MAP). A counterexample of order 3 is given, sho wing the answer is no, and that the conjecture fails ev en under the stronger condition of exact normalisation ( G 0 + G 1 ) 1 = 0 . The construction com bines a strictly p ositiv e exp onen tial baseline with a tw o-dimensional correction driven b y an irrational rotation. Strict p ositivit y of all joint densities follo ws from the contin uous-time damping of the correction blo c k; the obstruction to MAP realisability comes from the p oles of the b oundary generating function at e ± iφ , which cannot be p eripheral eigen v alues of any finite nonnegative matrix when φ/π / ∈ Q . 1 In tro duction Phase-t yp e (PH) distributions and matrix-exp onen tial (ME) distributions describ e the same kind of ob ject, a distribution on R + giv en by a vector–matrix pair ( ν , G 0 ) , but with different admissibilit y conditions . A PH represen tation comes from a finite Mark ov chain and is therefore sign constrained. An ME representation k eeps the same formula for the densit y , f ( t ) = − ν e G 0 t G 0 1 , but drops the Marko vian interpretation; the question of giving such a representation a prob- abilistic in terpretation go es bac k to Cox [ 6 ] and has b een revisited more recently [ 11 ]. The relation b et ween the t wo classes has b een studied for a long time. O’Cinneide’s theorem shows that strict p ositivit y together with a dominant-eigen v alue condition is enough to reco ver a PH represen tation from an ME one [ 10 ]. Later w ork made that picture more constructive [ 9 , 5 , 8 ]. The same distinction app ears for p oin t pro cesses. A finite-dimensional MAP is describ ed b y a triple ( ν , G 0 , G 1 ) with the usual Marko vian sign constrain ts: ν ≥ 0 , G 1 ≥ 0 , the off-diagonal en tries of G 0 are nonnegative, and G 0 1 ≤ 0 . A RAP , introduced by Asmussen and Bladt [ 1 ], k eeps the same joint-densit y form ula, f k ( t 1 , . . . , t k ) = ν e G 0 t 1 G 1 e G 0 t 2 G 1 · · · e G 0 t k G 1 1 , t 1 , . . . , t k ≥ 0 , (1) but allo ws arbitrary real entries as long as the resulting densities are nonnegativ e for every k . The relationship b etw een these t wo classes was survey ed in [ 2 ]; RAPs hav e also been used as mo delling tools in fluid queues [ 3 ] and quasi-birth-and-death processes [ 4 ]. Ev ery MAP is therefore a RAP , but not ev ery RAP is visibly Mark ovian. The real question is whether suitable regularit y ass umptions force the hidden Marko v structure back in. T elek formulated exactly that question in [ 12 ]. The conjecture is the p oint-process analogue of the classical ME/PH c haracterisation: if the joint densities are strictly p ositive and if G 0 has a unique real eigenv alue of maximal real part, does one automatically get a finite MAP represen tation? In the ME/PH setting the dominant-eigen v alue condition rules out the oscillatory b eha viour that preven ts a p ositive function from b eing PH. One might therefore exp ect the same 1 mec hanism to work here as well. The present note shows that it do es not. More than that, it still do es not work after one strengthens T elek’s admissibility requirements to the normalised setting of a non-terminating p oin t-pro cess mo del. The reason the ME/PH pro of do es not carry ov er is structural. In the one-matrix problem (that is, the c haracterisation of PH distributions), the densit y f ( t ) = − ν e G 0 t G 0 1 inv olv es only G 0 ; the sp ectral hypotheses on G 0 and the p ositivity of f together constrain the single matrix enough to force a Mark ovian realisation. In the tw o-matrix problem, the k -fold densit y ( 1 ) in volv es iterated pro ducts e G 0 t 1 G 1 e G 0 t 2 G 1 · · · , so the matrix G 1 en ters as an algebraically indep enden t ob ject. The h yp otheses of Conjecture 1 constrain G 0 directly (through the dominant-eigen v alue condition) and the pair ( G 0 , G 1 ) join tly (through p ositivit y of the densities), but they say almost nothing ab out G 1 alone. The question is therefore whether p ositivit y of all joint densities places enough indirect constraints on G 1 to force nonnegativit y . The answ er turns out to b e no: the con tinuous-time damping from G 0 can mask b ounded oscillatory behaviour in G 1 that is fundamen tally incompatible with any nonnegative realisation. Conjecture 1 (T elek [ 12 ]) . A ny RAP with strictly p ositive joint densities on (0 , ∞ ) k for every k ≥ 1 and with G 0 having a unique r e al eigenvalue of maximal r e al p art has a finite-dimensional MAP r epr esentation. T elek’s note formulates the problem with the w eaker condition Z ∞ 0 · · · Z ∞ 0 f k ( t 1 , . . . , t k ) dt 1 · · · dt k ≤ 1 . F or the present purp ose it is more natural to imp ose the exact consistency condition G 0 1 + G 1 1 = 0 , (2) whic h is the analogue of the usual conserv ation-of-mass iden tity in a MAP . This condition is implicit in the finite-dimensional p oin t-pro cess framew ork of Asmussen and Bladt [ 1 ], though it is not stated explicitly there; w e mak e it explicit here. Under ( 2 ) , together with ν 1 = 1 and e G 0 t 1 → 0 , each f k in tegrates to 1 and the family ( f k ) k ≥ 1 is pro jectiv ely consistent under in tegration in the last v ariable. The example b elo w is therefore not a defective or partially normalised ob ject; it is a gen uinely normalised RAP , and in particular it strengthens T elek’s setup. The counterexample w e pro vide has order 3 . The first co ordinate provides a strictly p ositiv e exp onen tial baseline against whic h a signed correction can b e added. The remaining tw o co ordinates form the correction blo c k, which decays faster in con tinuous time, with rate 2 . A t eac h arriv al, how ever, the correction blo ck is m ultiplied b y a rotation b y angle φ . Because orthogonal rotations ha ve b ounded p o wers, the p erturbation remains small and strict positivity is preserv ed. Cho osing φ/π irrational ensures that the successive rotation directions never rep eat p eriodically , whic h is precisely what a finite nonnegative matrix cannot repro duce on its p eripheral sp ectrum. Our result is th us conceptually close to the PH-c haracterisation literature, but with the opp osite conclusion. The classical results show that p ositivity together with the righ t spectral assumptions is enough to force a Marko vian representation. Here the same philosophy breaks do wn one lev el higher: p ositivity of all join t densities and a simple dominan t eigen v alue for G 0 still do not prev ent a non-Marko vian oscillatory mode from hiding in the arriv al mechanism. The heuristic that leads to the counterexample can b e describ ed as follows. If a MAP represen tation ( α , C 0 , C 1 ) were to exist, the b oundary v alues a k := ν G k 1 1 would hav e to equal αC k 1 1 for a nonnegative matrix C 1 . This is a scalar nonnegativ e realisation problem, and it carries a classical sp ectral obstruction: b y the Perron–F rob enius theorem, every eigenv alue of a finite nonnegative matrix on its sp ectral circle m ust b e a ro ot of unity times the sp ectral radius. The simplest b ounded oscillation that violates this constrain t is a rotation by an irrational angle: 2 the sequence e ij φ with φ/π / ∈ Q is b ounded (it liv es on the unit circle) but never p eriodic, since b y W eyl’s equidistribution theorem its v alues are dense on the circle. The generating function of the resulting b oundary sequence then has p oles at e ± iφ , which are not roots of unity . Once this obstruction is identified, the construction is almost forced: one needs a RAP whose matrix G 1 enco des an irrational rotation in a 2 × 2 blo c k, together with enough contin uous-time damping in G 0 to keep all join t densities strictly p ositiv e. The rest of the pap er is organised as follo ws. Section 2 presen ts the coun terexample in full and derives a closed form for the joint dens ities. Section 3 verifies exact normalisation, strict p ositivit y , and the dominan t-eigenv alue condition. Section 4 studies the b eha viour of the join t densities at the origin and computes the generating function of the resulting sequence. Section 5 uses these results to pro ve that no finite-dimensional MAP represen tation can exist. 2 The counterexample Throughout, matrices and v ectors are written in b old. W e write 1 for the all-on es column vector of appropriate dimension, 1 2 = (1 , 1) ⊤ for the all-ones column v ector in R 2 , 0 for the zero column vector of appropriate dimension, and I , I 2 for the iden tity matrix of appropriate and of size 2 , resp ectiv ely . The key idea is to encode inside the arriv al matrix G 1 a tw o-dimensional rotation acting on a correction part of the in tensity v ector; this pro duces the oscillatory terms in the b oundary sequence. The rotation is in tro duced via the matrix φ ∈ (0 , π ) , φ π / ∈ Q , (3) and c = cos φ, s = sin φ, R φ = c − s s c ! . A t each arriv al, the tw o correction coordinates of the in tensity vector are transformed by R φ . After j arriv als the correction comp onen t is multiplied by R j φ = R j φ , so each arriv al rotates the correction co ordinates b y one further angle φ . Since R φ is orthogonal, its p o wers remain b ounded. The condition φ/π / ∈ Q ensures that the successiv e directions never rep eat: the sequence { R j φ } j ≥ 0 is not even tually p erio dic. Since R φ has signed en tries it cannot serv e directly as a nonnegativ e arriv al matrix. A third co ordinate, the p ositiv e-intensit y coordinate, is therefore added; its sole role is to contribute enough p ositiv e mass to keep the total arriv al intensities nonnegative. A ccordingly , we set up G 1 in the blo c k form G 1 = b 0 ⊤ u R φ ! , u = u 1 u 2 ! , (4) where the b ottom-righ t block R φ acts on the correction co ordinates, the scalar b is the arriv al rate of the p ositiv e-intensit y co ordinate, and the vector u enco des the coupling b etw een that co ordinate and the rotating blo ck; all three are determined by the condition ( G 0 + G 1 ) 1 = 0 b elo w. W e tak e G 0 to b e diagonal, so that b et ween arriv als each co ordinate deca ys exponentially at its own rate with no further rotation or mixing. Cho osing a faster deca y rate for the correction co ordinates keeps their contribution small: G 0 =    − 1 0 0 0 − 2 0 0 0 − 2    . (5) The constants b , u 1 , u 2 in ( 4 ) are no w fixed b y ( 2 ), which gives b = 1 , u 1 = 2 − c + s, u 2 = 2 − c − s. (6) 3 Therefore G 1 =    1 0 0 2 − c + s c − s 2 − c − s s c    . (7) The initial distribution is ν = (1 − ε, ε, 0) , (8) where ε ∈ (0 , 1) is a small parameter; its admissible range is determined in Prop osition 4 . F or the closed-form joint densities, write T j = t 1 + · · · + t j , T = T k = t 1 + · · · + t k . Because G 0 is diagonal and G 1 is blo c k low er triangular, the pro duct in ( 1 ) can b e computed explicitly . Lemma 2. F or every k ≥ 1 and every t 1 , . . . , t k ≥ 0 , f k ( t 1 , . . . , t k ) = e − T   1 − ε + ε (1 , 0)   k − 1 X j =1 e − T j (2 R j − 1 φ − R j φ ) 1 2 + 2e − T R k − 1 φ 1 2     , (9) wher e 1 2 = (1 , 1) ⊤ is the al l-ones ve ctor in R 2 . Pr o of. Throughout this pro of, fix k 0 ≥ 1 . F or i = 1 , . . . , k 0 , set M i = e G 0 t i G 1 , so that f k 0 ( t 1 , . . . , t k 0 ) = ν M 1 · · · M k 0 1 . F or j = 1 , . . . , k 0 , write M j · · · M k 0 1 = e − ( T − T j − 1 ) v j ! , T 0 := 0 , with v j ∈ R 2 . Using the blo c k forms G 1 = 1 0 ⊤ u R φ ! , u = (2 I 2 − R φ ) 1 2 , and e G 0 t j = e − t j 0 ⊤ 0 e − 2 t j I 2 ! , w e obtain M j = e − t j 0 ⊤ e − 2 t j u e − 2 t j R φ ! . Hence v j = e − 2 t j u e − ( T − T j ) + e − 2 t j R φ v j +1 , j = 1 , . . . , k 0 − 1 . Also, since G 1 1 = − G 0 1 = (1 , 2 , 2) ⊺ , we hav e M k 0 1 = e G 0 t k 0 G 1 1 = e − t k 0 2e − 2 t k 0 1 2 ! , and therefore v k 0 = 2e − 2 t k 0 1 2 . Iterating the recursion giv es v 1 = k 0 − 1 X j =1 e − 2 T j R j − 1 φ u e − ( T − T j ) + 2e − 2 T R k 0 − 1 φ 1 2 . 4 Substituting u = (2 I 2 − R φ ) 1 2 and factoring out e − T yields v 1 = e − T   k 0 − 1 X j =1 e − T j (2 R j − 1 φ − R j φ ) 1 2 + 2e − T R k 0 − 1 φ 1 2   . Finally , since ν = (1 − ε, ε, 0) , w e ha ve f k 0 = (1 − ε )e − T + ε (1 , 0) v 1 , whic h is exactly ( 9 ). F orm ula ( 9 ) separates the densit y into a p ositiv e leading term (1 − ε )e − T and a signed correction. In the correction, each summand carries an extra factor e − T j , so although the oscillation in R j φ do es not deca y with j , its con tribution is damp ed by the elapsed time. 3 Normalisation, p ositivity , and the sp ectral condition W e v erify the three prop erties that qualify the example as a counterexample to Conjecture 1 . Prop osition 3. F or every k ≥ 2 , Z ∞ 0 f k ( t 1 , . . . , t k ) dt k = f k − 1 ( t 1 , . . . , t k − 1 ) , (10) and Z ∞ 0 · · · Z ∞ 0 f k ( t 1 , . . . , t k ) dt 1 · · · dt k = 1 for al l k ≥ 1 . (11) Pr o of. Since ( G 0 + G 1 ) 1 = 0 , we hav e Z ∞ 0 e G 0 t G 1 1 dt = − Z ∞ 0 e G 0 t G 0 1 dt = − h e G 0 t 1 i ∞ 0 = 1 . Substituting this into the last factor of ( 1 ) giv es ( 10 ) . Iterating ( 10 ) reduces ( 11 ) to the case k = 1 , which giv es R ∞ 0 f 1 ( t 1 ) dt 1 = ν 1 = 1 . Prop osition 4. Ther e is a c onstant M = M ( φ ) such that, whenever 0 < ε < 1 M + 1 , we have f k ( t 1 , . . . , t k ) > 0 for every k ≥ 1 and every t 1 , . . . , t k ≥ 0 . Pr o of. Fix k 0 ≥ 1 . By ( 9 ), f k 0 ( t 1 , . . . , t k 0 ) = e − T   1 − ε + ε   k 0 − 1 X j =1 e − T j (1 , 0)(2 R j − 1 φ − R j φ ) 1 2 + 2e − T (1 , 0) R k 0 − 1 φ 1 2     . Set w j := (1 , 0)(2 R j − 1 φ − R j φ ) 1 2 (1 ≤ j ≤ k 0 − 1) , w k 0 := 2(1 , 0) R k 0 − 1 φ 1 2 , and let W m := m X j =1 w j . 5 A direct telescoping computation giv es W m = (1 , 0)   2 m − 1 X j =0 R j φ − m X j =1 R j φ   1 2 = (1 , 0) I 2 − R m φ + m − 1 X ℓ =0 R ℓ φ ! 1 2 , 1 ≤ m ≤ k 0 − 1 , and W k 0 = W k 0 − 1 + 2(1 , 0) R k 0 − 1 φ 1 2 = (1 , 0)   I 2 + k 0 − 1 X ℓ =0 R ℓ φ   1 2 . Iden tify R 2 with C via ( x, y ) ↔ x + iy , so that R φ acts as multiplication by e iφ . Any real p olynomial A in R φ corresp onds to a complex scalar z A , and (1 , 0) A 1 2 = Re ( z A (1 + i )) , so | (1 , 0) A 1 2 | ≤ √ 2 | z A | . Sp ecifically , for A m := I 2 − R m φ + m − 1 X ℓ =0 R ℓ φ , w e hav e z A m = 1 − e imφ + m − 1 X ℓ =0 e iℓφ = 1 − e imφ + 1 − e imφ 1 − e iφ . Since φ ∈ (0 , π ) , we hav e e iφ  = 1 , so the denominator | 1 − e iφ | = 2 sin ( φ/ 2) is strictly p ositiv e (dropping the absolute v alue since sin( φ/ 2) > 0 ). By the triangle inequalit y , | z A m | ≤ | 1 − e imφ | + | 1 − e imφ | | 1 − e iφ | ≤ 2 + 2 2 sin( φ/ 2) = 2 + 1 sin( φ/ 2) . Therefore, | W m | ≤ √ 2 | z A m | ≤ √ 2  2 + 1 sin( φ/ 2)  . The same logic applies to W k 0 . Letting A k 0 := I 2 + P k 0 − 1 ℓ =0 R ℓ φ , its complex coun terpart z A k 0 giv es | W k 0 | ≤ √ 2 | z A k 0 | = √ 2      1 + 1 − e ik 0 φ 1 − e iφ      ≤ √ 2  1 + 1 sin( φ/ 2)  . Setting M := √ 2  2 + 1 sin( φ/ 2)  , hence | W m | ≤ M for all 1 ≤ m ≤ k 0 . No w write k 0 X j =1 e − T j w j = k 0 − 1 X j =1 W j  e − T j − e − T j +1  + W k 0 e − T k 0 . Since 0 ≤ e − T k 0 ≤ · · · ≤ e − T 1 ≤ 1 , it follo ws that       k 0 X j =1 e − T j w j       ≤ M k 0 − 1 X j =1 ( e − T j − e − T j +1 ) + M e − T k 0 = M e − T 1 ≤ M . Therefore f k 0 ( t 1 , . . . , t k 0 ) ≥ e − T  1 − ε − εM  , whic h is strictly p ositiv e as soon as ε < 1 / ( M + 1) . In the remainder of the paper, fix once and for all an ε satisfying 0 < ε < 1 / ( M + 1) . With this choice, the triple ( ν , G 0 , G 1 ) is indeed a RAP with strictly positive join t densities. Prop osition 5. The matrix G 0 has a unique r e al eigenvalue of maximal r e al p art, namely − 1 . Pr o of. The sp ectrum of G 0 is {− 1 , − 2 , − 2 } . So − 1 is simple and strictly dominates the other eigen v alues in real part. 6 4 Ev aluation at the origin and its generating function Setting t 1 = · · · = t k = 0 in ( 1 ) remo ves every factor e G 0 t i and leav es b ehind iterated p o wers of G 1 alone; define a k := ν G k 1 1 , k ≥ 1 . (12) If a MAP representation ( α , C 0 , C 1 ) exists, then a k = αC k 1 1 for a nonnegative matrix C 1 , so the question of MAP realisabilit y reduces, at the level of these v alues, to a nonnegative realisation problem for a scalar sequence, which is a classical problem with kno wn sp ectral obstructions. Lemma 6. F or every k ≥ 1 , a k = 1 + ε (1 , 0) k − 1 X j =0 R j φ 1 2 . (13) In p articular, ( a k ) k ≥ 1 is p ositive and b ounde d. Pr o of. Set t 1 = · · · = t k = 0 in ( 9 ). Then T j = 0 for all j , and therefore a k = 1 − ε + ε (1 , 0)   k − 1 X j =1 (2 R j − 1 φ − R j φ ) 1 2 + 2 R k − 1 φ 1 2   . The sum telescop es: k − 1 X j =1 (2 R j − 1 φ − R j φ ) + 2 R k − 1 φ = I 2 + k − 1 X j =0 R j φ . Hence a k = 1 − ε + ε (1 , 0)   I 2 + k − 1 X j =0 R j φ   1 2 = 1 + ε (1 , 0) k − 1 X j =0 R j φ 1 2 , whic h is ( 13 ) . The sequence is p ositiv e because a k = f k (0 , . . . , 0) > 0 by Prop osition 4 . F or b oundedness, note that under the iden tification of R 2 with C , (1 , 0) k − 1 X j =0 R j φ 1 2 is identified with Re k − 1 X j =0 e ij φ (1 + i ) = Re 1 − e ikφ 1 − e iφ (1 + i ) ! , whose mo dulus is b ounded by      1 − e ikφ 1 − e iφ (1 + i )      = | 1 − e ikφ | | 1 − e iφ | √ 2 ≤ 2 2 sin( φ/ 2) √ 2 = √ 2 sin( φ/ 2) , uniformly in k . Hence ( a k ) k ≥ 1 is b ounded. Lemma 7. The gener ating function A ( z ) := ∞ X k =1 a k z k is given by A ( z ) = z 1 − z  1 + ε 1 − z (cos φ + sin φ ) 1 − 2 z cos φ + z 2  . (14) Its only p ossible p oles ar e z = 1 and z = e ± iφ , and the p oles at z = e ± iφ do not c anc el. 7 Pr o of. By ( 13 ), A ( z ) = ∞ X k =1 z k + ε (1 , 0) ∞ X k =1 z k k − 1 X j =0 R j φ 1 2 = z 1 − z + εz 1 − z (1 , 0) ∞ X j =0 ( z R φ ) j 1 2 = z 1 − z + εz 1 − z (1 , 0)( I 2 − z R φ ) − 1 1 2 , where the interc hange of summation in the second step uses | z | < 1 , and the resulting rational expression ( 14 ) is understo od as the analytic con tinuation of A to C min us its poles. A direct calculation gives (1 , 0)( I 2 − z R φ ) − 1 1 2 = 1 1 − 2 z cos φ + z 2 (1 , 0) 1 − z cos φ − z sin φ z sin φ 1 − z cos φ ! 1 2 = 1 − z (cos φ + sin φ ) 1 − 2 z cos φ + z 2 , whic h pro ves ( 14 ). The denominator factors as 1 − 2 z cos φ + z 2 = (1 − z e iφ )(1 − z e − iφ ) , so, b esides the p ossible p ole at z = 1 , the only other p ossible p oles are at z = e ± iφ . Since we ha ve fixed ε > 0 , these p oles can disapp ear only if the numerator 1 − z ( cos φ + sin φ ) v anishes there. A t z = e iφ , this would require 1 = e iφ (cos φ + sin φ ) . T aking imaginary parts gives 0 = (cos φ + sin φ ) sin φ. Because φ ∈ (0 , π ) , we hav e sin φ > 0 , hence cos φ + sin φ = 0 . Substituting this back in to the previous equation yields 1 = 0 , a contradiction. Thus the numerator do es not v anish at z = e iφ . The case z = e − iφ is identical. 5 Non-existence of a MAP represen tation The argument uses only the sequence ( a k ) and the following prop osition, wh ose pro of rests on P erron–F rob enius theory for nonnegativ e matrices, applied blo c kwise via the F rob enius normal form. An eigenv alue of a matrix is called peripheral if its mo dulus equals the spectral radius. F or an irreducible nonnegativ e matrix, the Perron–F rob enius theorem implies that the p eripheral eigen v alues are alwa ys ro ots of unity times the spectral radius, meaning that its discrete-time ev olution can p erm ute mass among states cyclically but only with a finite p erio d. F or a reducible matrix this applies to each irreducible diagonal blo ck separately . In con trast, a real matrix with no sign constrain ts can enco de a rotation by an arbitrary angle, pro ducing a b ounded oscillation with irrational frequency . The RAP framework allows G 1 to carry an aperio dic rotation, while the MAP framew ork forces C 1 ≥ 0 , restricting its p eripheral sp ectrum to finitely man y equally-spaced phases. Prop osition 8. L et C ≥ 0 b e a finite squar e matrix and α ≥ 0 a r ow ve ctor, and supp ose that b k = αC k 1 ( k ≥ 1) is a b ounde d se quenc e. L et B ( z ) := P ∞ k =1 b k z k and supp ose that B has a p ole at some z 0 with | z 0 | = 1 . Then: 8 (i) every p ole of B is the r e cipr o c al of a nonzer o eigenvalue of C ; (ii) ρ ( C ) = 1 and C has an eigenvalue λ = z − 1 0 with | λ | = 1 ; (iii) every such eigenvalue λ is a r o ot of unity. Pr o of. By discarding states unreachable from the supp ort of α , we ma y assume without loss of generalit y that all states of C are reachable; this do es not affect ( b k ) or B . The identit y B ( z ) = α z C ( I − z C ) − 1 1 , v alid first for | z | < 1 /ρ ( C ) and then by analytic con tinuation, shows that ev ery p ole of B is the recipro cal of a nonzero eigenv alue of C (since the p oles corresp ond to roots of det ( I − z C ) ). If B has a p ole at z 0 with | z 0 | = 1 , then C has an eigenv alue λ = z − 1 0 with | λ | = 1 , so ρ ( C ) ≥ 1 . W e sho w ρ ( C ) ≤ 1 . Assume for con tradiction that ρ ( C ) > 1 . Put C in to F rob enius normal form and c ho ose an irreducible diagonal blo c k D with sp ectral radius ρ ( C ) . Since all states are reachable from the supp ort of α , there is some m suc h that β := ( αC m ) D is a nonzero nonnegativ e row vector. By [ 7 , Theorem 8.4.4(c)], D has a p ositiv e right eigenv ector q with D q = ρ ( C ) q with en tries that sum 1 ; in particular q ≤ 1 D en trywise. Since C ≥ 0 is blo c k upp er-triangular in F rob enius normal form, ( C n ) D D = D n and ( αC m + n ) D ≥ β D n . Therefore b m + n = αC m + n 1 ≥ β D n 1 D ≥ β D n q = ( β q ) ρ ( C ) n . Since β q > 0 , the right-hand side is unbounded, contradicting the b oundedness of ( b k ) . Thus ρ ( C ) = 1 . Finally , let λ ∈ σ ( C ) satisfy | λ | = 1 = ρ ( C ) . By F rob enius normal form, λ b elongs to some irreducible diagonal blo c k D with ρ ( D ) = ρ ( C ) = 1 , and Perron–F rob enius implies that ev ery eigen v alue of D on the unit circle is of the form e 2 π ir/h , 0 ≤ r ≤ h − 1 , for some integer h ≥ 1 [ 7 , Corollary 8.4.6(c)]. Thus λ is a root of unity . Theorem 9. F or the fixe d choic e of ε satisfying Pr op osition 4 , the RAP define d by ( 8 ) , ( 7 ) , and ( 5 ) has no e quivalent finite-dimensional MAP r epr esentation. Pr o of. Assume that an equiv alen t finite-dimensional MAP represen tation ( α , C 0 , C 1 ) exists, so in particular α ≥ 0 and C 1 ≥ 0 . Since b oth matrix-exp onential density formulas are con tinuous on [0 , ∞ ) k and agree on (0 , ∞ ) k , they also agree at t 1 = · · · = t k = 0 . Therefore setting t 1 = · · · = t k = 0 in b oth formulas gives b k := αC k 1 1 satisfying b k = a k for all k ≥ 1 , so B ( z ) = A ( z ) . By Prop osition 8 , ev ery p ole of B is the recipro cal of an eigenv alue of C 1 . By Lemma 7 , A ( z ) has p oles at z = e ± iφ , hence so do es B ( z ) . Therefore C 1 has eigenv alues λ ± = e ∓ iφ . Since | λ ± | = 1 , Prop osition 8 implies that λ ± m ust b e ro ots of unity . But e ± iφ are not ro ots of unity b ecause φ/π / ∈ Q . This contradiction prov es that no equiv alen t finite-dimensional MAP representation exists. The example shows that the hypotheses sufficient for the one-matrix ME/PH problem do not carry ov er to the tw o-matrix RAP/MAP problem: p ositivity of all joint densities and a simple dominan t eigenv alue for G 0 do not control the p eripheral b eha viour of the arriv al matrix, since the irrational rotation pro duces b oundedness without p erio dicit y , the generating function inherits the irrational phases as p oles, and Perron–F rob enius forces an y nonnegative matrix to ha ve only ro ot-of-unit y phases on its sp ectral circle. The characterisation of MAPs within the class of RAPs is lik ely a harder problem than the already difficult characterisation of PH distributions within ME distributions, whic h w as op en for sev eral decades b efore b eing resolved b y O’Cinneide [ 10 ]. In the one-matrix problem, all sp ectral information is concentrated in G 0 , and the dominan t-eigenv alue condition together with p ositivit y 9 suffices. 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