Gamma-Dirichlet Structure and Two Classes of Measure-valued Processes
aa r X i v : . [ m a t h . P R ] D ec Gamma-Dirichlet Structure and Two Classesof Measure-Valued Processes ∗ Shui Feng and Fang XuMcMaster UniversityMay 21, 2018
Abstract
The Gamma-Dirichlet structure corresponds to the decomposition of the gammaprocess into the independent product of a gamma random variable and a Dirichletprocess. This structure allows us to study the properties of the Dirichlet processthrough the gamma process and vice versa. In this article, we begin with a briefreview of existing results concerning the Gamma-Dirichlet structure. New results areobtained for the large deviations of the jump sizes of the gamma process and the quasi-invariance of the two-parameter Poisson-Dirichlet distribution. The laws of the gammaprocess and the Dirichlet process are the respective reversible measures of the measure-valued branching diffusion with immigration and the Fleming-Viot process with parentindependent mutation. We view the relation between these two classes of measure-valued processes as the dynamical Gamma-Dirichlet structure. Other results of thisarticle include the derivation of the transition function of the Fleming-Viot processwith parent independent mutation from the transition function of the measure-valuedbranching diffusion with immigration, and the establishment of the reversibility of thelatter. One of these is related to an open problem by Ethier and Griffiths and the otherleads to an alternative proof of the reversibility of the Fleming-Viot process.
Keywords: branching process with immigration, coalescent, Dirichlet process, gammaprocess, Hamiltonian, large deviations, quasi-invariant, reversibility, random time-change.
AMS 2001 subject classifications:
Primary 60F10; secondary 92D10. ∗ Research supported by the Natural Science and Engineering Research Council of Canada Introduction
Recall that for any α > , β >
0, the
Gamma ( α, β ) distribution has the density function f ( x ) = 1Γ( α ) β α x α − e − xβ , x > , and the Laplace transform Z ∞ e − ux f ( x ) d x = exp {− α log(1 + βu ) } , u > − /β, where α is the shape parameter and β is the scale parameter. A characterization of thegamma distribution obtained in [12] states that two independent positive random variables Y and Y are gamma random variables with the same scale parameter if and only if Y + Y and Y Y + Y are independent.For i = 1 , , let Y i be a Gamma ( α i , β ) random variable with α i > Y and Y are independent. Then we have( Y , Y ) = ( Y + Y )( Y Y + Y , Y Y + Y ) (1.1)= radial part × angular part , where Y + Y and ( Y Y + Y , Y Y + Y ) are independent with respective distributions Gamma ( α + α , β ) and Beta ( α , α ). The equation (1 .
1) is called the one-dimensional Gamma-Dirichletstructure .Let S be a compact metric space, ν a probability on S , θ and β any two positive numbers.The space of all non-negative finite measures on S , denoted by M ( S ), is equipped with theweak topology, and M ( S ), a subspace of M ( S ), consists of all probability measures on S .The integration of a measurable function g on S with respect to a measure µ in M ( S ) isdenoted by h µ, g i . We denote by B ( S ) and C ( S ) the respective sets of bounded measurablefunctions and continuous functions on S . The gamma process with shape parameter θν andscale parameter β is given by Y βθ,ν ( · ) = β ∞ X i =1 γ i ( θ ) δ ξ i ( · )where γ ( θ ) > γ ( θ ) > · · · are the points of the inhomogeneous Poisson point process on(0 , ∞ ) with mean measure θx − e − x d x , and independently, ξ , ξ , . . . are i.i.d. with commondistribution ν .Denote the law of Y βθ,ν by Γ βθ,ν . The corresponding Laplace functional is2 M ( S ) e −h µ,g i Γ βθ,ν ( d µ ) = exp {− θ h ν , log(1 + βg ) i} where g ( s ) > − /β, for all s ∈ S. Set σ ( θ ) = ∞ X i =1 γ i ( θ ) , P i ( θ ) = γ i ( θ ) σ ( θ ) , X θ,ν ( · ) = ∞ X i =1 P i ( θ ) δ ξ i ( · ) . The law of ( P ( θ ) , P ( θ ) , . . . ), denoted by P D ( θ ), is the Poisson-Dirichlet distribution with parameter θ (cf. [13]), X θ,ν ( · ) is the Dirichlet process with law denoted by Π θ,ν , andthe relation X θ,ν ( · ) = Y βθ,ν ( · ) Y βθ,ν ( S ) (1.2)is called the infinite-dimensional Gamma-Dirichlet structure .The law Γ βθ,ν is the reversible measure of a measure-valued branching diffusion withimmigration (henceforth MBI) with generator L = 12 {h µ, δ δµ ( s ) i + h θν − λµ, δδµ ( s ) i} (1.3)where λ = β > δϕδµ ( s ) = lim ǫ → ϕ ( µ + ǫδ s ) − ϕ ( µ ) ǫ . The law Π θ,ν is the reversible measure of the Fleming-Viot process with parent independentmutation (henceforth FVP) with generator A = 12 {h ν ( d s ) · ( δ s − ν )( d s ) , δ δν ( s ) δν ( s ) i + θ h ν − ν, δδν ( s ) i} . (1.4)The dynamical Gamma-Dirichlet structure corresponds to the relation between these twoclasses of measure-valued processes.Section 2 reviews some known results including several algebraic identities, the formalHamiltonian, the quasi-invariant, and large deviations. The large deviation principle (hence-forth LDP) for the jump sizes of the gamma process is established in Section 3. Section 4obtains new results on quasi-invariance for the jump size of gamma process and the two-parameter Poisson-Dirichlet distribution. Finally in section 5, we derive the transition func-tion of the FVP process directly from the transition function of the MBI process througha time change, and establish the reversibility of the MBI process. This, combined with theGamma-Dirichlet structure, provides an alternative proof of the reversibility of the FVPprocess. 3 Gamma-Dirichlet Structure
The gamma distribution is structurally similar to the normal distribution in many ways.For example, a normal population is characterized by the independency of any translation-invariant statistic and the sample mean while a gamma population is characterized by theindependency of any scale-invariant statistic of the sample mean or equivalently by the one-dimensional Gamma-Dirichlet structure. In this section, we collect several existing resultsassociated with the Gamma-Dirichlet structure, which motivate the studies in subsequentsections.
Let d = denote equality in distribution. For any integer m ≥ β > θ i > ν i ∈ M ( S ) ,i = 1 , . . . , m , let Y βθ ,ν , . . . , Y βθ m ,ν m be m independent gamma processes with the same scaleparameter β . By direct calculation, we obtain the following additive property: Y βθ ,ν + · · · + Y βθ m ,ν m d = Y βθ + ··· + θ m , θ θ ··· + θm ν + ··· + θmθ ··· + θm ν m . (2.1)Let X θ ,ν , . . . , X θ m ,ν m be m independent Dirichlet processes, and independently let therandom vector ( η , . . . , η m ) have a Dirichlet ( θ , . . . , θ m ) distribution. Then the followingmixing identity follows from (2 .
1) and the infinite-dimensional Gamma-Dirichlet structure. η X θ ,ν + · · · + η m X θ m ,ν m d = X θ + ··· + θ m , θ θ ··· + θm ν + ··· + θmθ ··· + θm ν m . (2.2)Let U ( θ ) , U ( θ ) , . . . be a sequence of i.i.d. random variables with common distribution Beta (1 , θ ). Set V ( θ ) = U ( θ ) , V n ( θ ) = U n ( θ ) n − Y i =1 (1 − U i ( θ )) , n ≥ . (2.3)Then the law of ( V ( θ ) , V ( θ ) , . . . ) is the GEM distribution with parameter θ and the lawof the descending order statistics of ( V ( θ ) , V ( θ ) , . . . ) is the Poisson-Dirichlet distribution P D ( θ ). The Dirichlet process X θ,ν is then given by X θ,ν = ∞ X i =1 V i ( θ ) δ ξ i , where ( V ( θ ) , V ( θ ) , . . . ) is independent of ( ξ , ξ , . . . ). It follows from the self-similarity ofthe GEM representation (2 .
3) that X θ,ν = V ( θ ) δ ξ + (1 − V ( θ )) ˜ X θ,ν , (2.4)where ˜ X θ,ν is an independent copy of X θ,ν . 4 .2 Quasi-Invariance Consider a measure P on space S . Let G denote a group of transformations from S to S .The measure P is quasi-invariant with respect to the group G if for any T in G the imagemeasure P ◦ T − of P under T is equivalent to P . In this case, T preserves zero sets. If P ◦ T − = P , the measure P is invariant with respect to G .The quasi-invariant property of Gaussian measure plays a major role in stochastic cal-culus. In [18] and [19], the quasi-invariance of the gamma process is studied thoroughly. Acomparison with the Gaussian measure reveals a remarkable similarity between the sphericalsymmetry of the Gaussian measure and the multiplicative symmetry of the gamma process.This provides a different aspect for the Gamma-Dirichlet structure.Let B + ( S ) be the collection of positive Borel measurable functions on S with strictlypositive lower bound. For each f in B + ( S ), set T f ( µ )( d s ) = f ( s ) µ ( d s ) , µ ∈ M ( S ) , and let T f (Γ βθ,ν ) denote the image law of Γ βθ,ν under T f . Then the following holds. Theorem 2.1 (Tsilevich and Vershik [18], Tsilevich, Vershik and Yor [19]) . For any f in B + ( S ) , T f (Γ βθ,ν ) and Γ βθ,ν are mutually absolutely continuous and d T f (Γ βθ,ν ) d Γ βθ,ν ( µ ) = exp {− [ θ h ν , log f i + h µ, β − ( f − − } . (2.5)For each f in B + ( S ), set T f ( ν )( d s ) = f ( s ) ν ( d s ) h ν, f i , ν ∈ M ( S ) , and denote by T f (Π θ,ν ) the image law of Π θ,ν under T f . Theorem 2.2 (Handa [9])
The laws T f (Π θ,ν ) and Π θ,ν are mutually absolutely continuousand d T f (Π θ,ν ) d Π θ,ν ( ν ) = exp {− θ [ h ν , log f i + log h ν, f − i ] } . (2.6)Let ∇ ∞ = ( p = ( p , p , . . . ) : p ≥ p ≥ · · · ≥ , ∞ X j =1 p j = 1 ) be equipped with the subspace topology of the infinite-dimensional Euclidean space. Thenthe Poisson-Dirichlet distribution P D ( θ ) is a probability measure on ∇ ∞ . For any f in5 + ( S ) and any i.i.d. sequence { ξ i : i = 1 , , . . . } with common distribution ν , define themap S f,ν : ∇ ∞ → ∇ ∞ , ( p , p , . . . ) (˜ p , ˜ p , . . . ) (2.7)where (˜ p , ˜ p , . . . ) is the descending order statistics of (cid:18) f ( ξ ) p P ∞ i =1 f ( ξ i ) p i , f ( ξ ) p P ∞ i =1 f ( ξ i ) p i , . . . (cid:19) . Theorem 2.3 ( Tsilevich and Vershik [18])
Let g P D ( θ ) denote the image law of P D ( θ ) underthe map S f,ν . Then g P D ( θ ) and P D ( θ ) are mutually absolutely continuous and d g P D ( θ ) d P D ( θ )( p ) = exp {− θ h ν , log f i} · Z ∞ u θ − Γ( θ ) ∞ Y i =1 h ν , e − uf − ( ξ i ) p i i d u. (2.8)Consider an abstract space Ω with a formal reference probability measure P (uniformor invariant in some sense). The formal Hamiltonian H ( ω ) is a function associated withanother probability Q such that Q ( dω ) = Z − exp {−H ( ω ) }P ( d ω ) . For each µ ∈ M ( S ) \ { } , set ˆ µ ( · ) = µ ( · ) µ ( S ) ∈ M ( S ). Let φ ( x ) = x log x − ( x − , x ≥ , and for any ν , ν ∈ M ( S ) Ent ( ν | ν ) = ( R M ( S ) log d ν d ν d ν , ν ≪ ν + ∞ , else . Then the following holds.
Theorem 2.4 (Handa [9])
The formal Hamiltonian for the Gamma process Γ βθ,ν is given by H g ( µ ) = θEnt ( ν | ˆ µ ) + µ ( S ) β φ ( βθµ ( S ) ) (2.9)= angular component + radial component , and the formal Hamiltonian for the Dirichlet process Π θ,ν is given by H d ( ν ) = θEnt ( ν | ν ) , ν ∈ M ( S ) . (2.10) Remark
For βθ = 1, we have H g ( µ ) | µ ( S )=1 = H d (ˆ µ )which, combined with (2 . Asymptotic Behavior for the Jump Sizes of theGamma Process
The Gamma-Dirichlet structure has been used in [8] and [7] to obtain the fluctuation the-orem for the Poisson-Dirichlet distribution. In this section, we apply the Gamma-Dirichletstructure to the establishment of the LDP for the jump sizes of the Gamma process.Large deviations for a family of probability measures { P λ : λ ∈ index set } on a space E are estimations of the following type: P λ { G } ≍ exp {− a ( λ ) inf x ∈ G I ( x ) } , where a ( λ ) is called the large deviation speed, and the nonnegative lower semi-continuousfunction I ( · ) is called the rate function. The rate function is good if { x ∈ E : I ( x ) ≤ c } is compact for any nonnegative constant c . Additional terminologies and general results onLDPs are found in [2].Recall that γ ( θ ) > γ ( θ ) > · · · are the points of the inhomogeneous Poisson point processon (0 , ∞ ) with mean measure θx − e − x d x , and( γ ( θ ) , γ ( θ ) , . . . ) = σ ( θ )( P ( θ ) , P ( θ ) , . . . ) (3.1)where σ ( θ ) is a Gamma ( θ,
1) random variable that is independent of the Poisson-Dirichletdistributed random sequence ( P ( θ ) , P ( θ ) , . . . ). Theorem 3.1
Let P θ denote the law of θ − ( γ ( θ ) , γ ( θ ) , . . . ) on R ∞ + . Then the family { P θ : θ > } satisfies a LDP with speed θ and good rate function I ( x , x , . . . ) = ∞ X i =1 x i , x ≥ x ≥ · · · ≥ ∞ , otherwise , as θ converges to infinity. Proof:
It follows from direct calculation that the laws of the family { θ − σ ( θ ) : θ > } satisfya LDP as θ tends to infinity with speed θ and good rate function I ( y ) = ( y − − log y, y > ∞ , otherwise , Let ∇ = { ( p , ..., p n , ... ) : p ≥ p ≥ · · · ≥ , ∞ X i =1 p i ≤ } .
7t follows from Theorem 4.4 in [1] that the family { P D ( θ ) : θ > } satisfies a LDP onspace ∇ as θ tends to infinity with speed θ and good rate function I ( p , p , . . . ) = − log(1 − ∞ X i =1 p i ) , P ∞ i =1 p i < ∞ , otherwise , The Gamma-Dirichlet structure (3 . { P θ : θ > } satisfies a LDP as θ tends to infinity with speed θ and goodrate function inf { I ( y ) + I ( p , p , . . . ) : y ≥ , ( p , . . . ) ∈ ∇ , x i = yp i , i = 1 , . . . } which equals to I ( x , x , . . . ). ✷ Remarks
The LDP is established in [6] for
P D ( θ ) when θ converges to zero. The speed is − log( θ ) and the good rate function is I ( p , p , . . . ) = , p = 1 n − , P ni =1 p i = 1 , p n > ∞ , else . Since ( γ ( θ ) , γ ( θ ) , . . . ) converges to (0 , , . . . ) when θ converges to zero, one would liketo establish the LDP for ( γ ( θ ) , γ ( θ ) , . . . ) from the LDP for P D ( θ ). A direct calculationshows that a LDP holds for σ ( θ ) with speed − log( θ ) and rate function I ( y ) = ( , y = 01 , otherwise , Since I ( · ) is not a good rate function, the contraction principle can not be applied in thiscase. On the other hand, for any r ≥ γ ( θ ) , . . . , γ r ( θ )) is θ r x · · · x r exp {− r X i =1 x i − θE ( x r ) } , x ≥ · · · ≥ x r > E ( x ) = Z ∞ x z − e − z d z. By direct calculation, one can show that a LDP holds for ( γ ( θ ) , γ ( θ ) , . . . ) with speed − log( θ ) and rate function I ( x , x , . . . ) = , x i = 0 , i ≥ n, x ≥ · · · ≥ x n > , x m = 0 , m ≥ n + 1 ∞ , otherwise , I ( · ) is not a good rate function. But remarkably the three rate functions have arelation that is consistent with the contraction principle, i.e., I ( x , x , . . . ) = inf { I ( p , . . . ) + I ( y ) : yp i = x i , i = 1 , . . . } . For any 0 < α < θ > − α , let U k ( α, θ ) , k = 1 , , ... , be a sequence of independentrandom variables such that U i ( α, θ ) has Beta (1 − α, θ + iα ) distribution. Set V ( α, θ ) = U ( α, θ ) , V n ( α, θ ) = (1 − U ( α, θ )) · · · (1 − U n − ( α, θ )) U n ( α, θ ) , n ≥ . (4.1)Then the law of ( V ( α, θ ) , V ( α, θ ) , ... ) is called the two-parameter GEM distribution, denotedby GEM ( θ, α ) . Let P ( α, θ ) = ( P ( α, θ ) , P ( α, θ ) , ... ) denote ( V ( α, θ ) , V ( α, θ ) , ... ) in descend-ing order. The law of P ( α, θ ) is called the two-parameter Poisson-Dirichlet distribution, and,following [15], is denoted by P D ( α, θ ). In this section, we will generalize Theorem 2.1 andTheorem 2.3 to the jump sizes of the gamma process and the two-parameter Poisson-Dirichletdistribution, respectively.Let M a ( S ) = { ∞ X i =1 x i δ ξ i : ξ i ∈ S, x i ≥ , < ∞ X i =1 x i < ∞} denote the space of non-negative finite atomic measures on S , equipped with the subspacetopology of M ( S ). Let R ↓ + = { x = ( x , x , . . . ) : x ≥ x ≥ · · · ≥ , ∞ X i =1 x i < ∞} , and for any ν = P ∞ i =1 y i δ ξ i in M a ( S ) define the map J : M a ( S ) → R ↓ + , ν x , where x is ( y , y , . . . ) in descending order. Similarly we define the map T : M a ( S ) → ∇ ∞ , ν p = ( p , p , . . . )where p is { y i P ∞ j =1 y j : i = 1 , , . . . } in descending order. Since the usual weak topologygenerates the same Borel σ -field on M a ( S ) as the weak atomic topology, it follows fromLemma 2.5 in [5] that the maps J and T are measurable.9 heorem 4.1 For any f in B + ( S ) , let ˜ T f (Γ βθ,ν ) and ˜Γ βθ,ν be the respective image laws of T f (Γ βθ,ν ) and Γ βθ,ν under J . Then ˜ T f (Γ βθ,ν ) and ˜Γ βθ,ν are mutually absolutely continuousand d ˜ T f (Γ βθ,ν ) d ˜Γ βθ,ν ( x ) = exp {− θ h ν , log f i} E ν ∞ [exp {− ∞ X i =1 β − ( f − ( ξ i ) − x i } ] . (4.2) Proof:
For any bounded measurable function F on R ↓ + , it follows from Theorem 2.1 that E ˜ T f (Γ βθ,ν ) [ F ( x )] = E T f (Γ βθ,ν ) [ F ( J ( µ ))]= exp {− θ h ν , log f i} E Γ βθ,ν [ F ( J ( µ )) exp {−h µ, β − ( f − − } ]= exp {− θ h ν , log f i} E ˜Γ βθ,ν " F ( x ) E ν ∞ [exp {− ∞ X i =1 β − ( f − ( ξ i ) − x i } ] which leads to the result. ✷ The stable subordinator with index α is a L´evy process with L´evy measure d Λ α = cα Γ(1 − α ) s − α − d s, s > c >
0. Let ρ ( α ) ≥ ρ ( α ) ≥ . . . denote the descending order jump sizes of the stablesubordinator over the interval (0 , ξ , ξ , . . . be i.i.d. with commondistribution ν . Then the ν -scaled stable subordinator has the form ∞ X i =1 ρ i ( α ) δ ξ i with law denoted by P c,ν α, . The Laplace functional of P c,ν α, is given by E P c,ν α, [exp {−h ν, g i} ] = exp {− c h ν , g α i} . Lemma 4.2
For any f in B + ( S ) , let A α = h ν , f α i and ν α ( d s ) = f α ( s ) ν ( d s ) /A α . Thenthe law T f ( P c,ν α, ) is P cA α ,ν α α, . Proof:
For any g in B + ( S ), it follows from direct calculation that E T f ( P c,ν α, ) [exp {−h ν, g i} ] = E P c,ν α, [exp {−h ν, gf i} ]= exp {− c h ν , f α g α i} = exp (cid:26) − cA α h f α ν A α , g α , i (cid:27) = E P cAα,να α, [exp {−h ν, g i} ] , ✷ Next set c α,θ = c θ/α Γ( θ +1)Γ( θ/α +1) and consider the law P c,ν α,θ defined by P c,ν α,θ ( d ν ) = c α,θ ν ( S ) θ P c,ν α, ( d ν ) . (4.3)It is known (cf. [14], [15]) that T ( P c,ν α, ) = P D ( α,
0) (4.4)and T ( P c,ν α,θ ) = P D ( α, θ ) . (4.5)The next theorem generalizes the result in Theorem 2.3 to the two-parameter Poisson-Dirichlet distribution P D ( α, θ ). Theorem 4.3
For any f in B + ( S ) , let g P D ( α, θ ) denote the image law of P D ( α, θ ) under themap S f,ν defined in (2 . . Then g P D ( α, θ ) and P D ( α, θ ) are mutually absolutely continuousand d g P D ( α, θ ) d P D ( α, θ )( p ) = h ν , f α i − θ/α θ ) Z ∞ u θ − ∞ Y i =1 E ν α [ e − uf − ( ξ i ) p i ] d u. (4.6) Proof:
For any nonnegative product measurable function Φ on ∇ ∞ , we have E g P D ( α,θ ) [Φ( p )] = E T f ( P c,ν α,θ ) [Φ( T ( µ ))]= E P c,ν α,θ [Φ( T ( T f ( ν ))]= E P c,ν α, (cid:20) Φ( T ( T f ( ν )) c α,θ ν ( S ) θ (cid:21) . Since ν ( S ) can be written as h T f ( ν ) , f − i , it follows from lemma 4.2 that E P c,ν α, (cid:20) Φ( T ( T f ( ν )) c α,θ ν ( S ) θ (cid:21) = E P c,ν α, (cid:20) Φ( T ( T f ( ν )) c α,θ h T f ( ν ) , f − i θ (cid:21) = E T f ( P c,ν α, ) (cid:20) Φ( T ( µ )) c α,θ h µ, f − i θ (cid:21) = E T f ( P cAα,ναα, ) (cid:20) Φ( T ( µ )) c α,θ h µ, f − i θ (cid:21) = E P cAα,ναα,θ (cid:20) Φ( T ( µ )) c α,θ ( cA α ) α,θ µ ( S ) θ h µ, f − i θ (cid:21) = E P D ( α,θ ) " Φ( p ) E ν ∞ α [ 1 A θ/αα ( P f − ( ξ i ) p i ) θ ] , ν ∞ α is the infinite product of ν α . Hence we have g P D ( α, θ ) P D ( α, θ )( d p ) = E ν ∞ α [ 1 A θ/αα ( P f − ( ξ i ) p i ) θ ] , Note that for λ > λ − θ = R ∞ σ θ − Γ( θ ) e − λσ dσ . Consequently, E ν ∞ α (cid:20) P f − ( ξ i ) p i ) θ (cid:21) = E ν ∞ α (cid:20)Z ∞ σ θ − Γ( θ ) e − σ P f − ( ξ i ) p i dσ (cid:21) = 1Γ( θ ) Z ∞ σ θ − ∞ Y i =1 E ν α [ e − σf − ( ξ i ) p i ] dσ, which leads to (4 . f has a strictly positive lower bound implies that ∞ Y i =1 E ν α [ e − σf − ( ξ i ) p i ]is strictly positive. Therefore the right hand side of (4 .
6) is finite everywhere. ✷ Remarks
When θ = 0, the density equals to one and P α, is a fixed point for the map S f,ν .On the other hand, by direct calculation it follows that ν α ⇒ ν , α → α → h ν , f α i − θ/α = exp {− θ h ν , log f i} . Therefore, for positive θ , the quasi-invariance of P D ( θ ) can be obtained from the quasi-invariance of P D ( α, θ ) by taking the limit of α going to zero. Let Y t denote the MBI process with generator L given by (1 . X t denotes theFVP process with generator A given by (1 . Y t and X t . For λ ≥ , t ≥
0, set C ( λ, t ) = ( λ − (e λt/ − , λ = 0 ,t/ , λ = 0 . (5.1)12or any a >
0, let { N a ( t ) : t ≥ } be a time inhomogeneous pure death Markov chain withdeath rate N a ( t ) / C ( − λ, t ) at time t >
0, entrance boundary ∞ , and marginal distribution q a,λn ( t ) = P { N a ( t ) = n } = a n C n ( λ, t ) n ! exp {− a/C ( λ, t ) } , n = 0 , , . . . . Let Z + = { , , . . . } and ˆ Z + denote the one-point compactification of Z + by ∞ . For any θ >
0, let { D θt : t ≥ } be the embedded chain of Kingman’s coalescent. It is a pure deathprocess with state space ˆ Z + , death rates λ n = n ( n + θ − , n = 0 , , . . . , and entrance boundary ∞ . For t >
0, let d θn ( t ) = P ( D θt = n )denote the probability of having n lines of decent at time t beginning at generation zero.The following explicit formula for d θn ( t ) is obtained in [17]. d θn ( t ) = − ∞ P m =1 (2 m − θ )( m !) − ( − m − θ ( m − e − λ m t , n = 0 ∞ P m = n (2 m − θ )( m !) − ( − m − n (cid:0) mn (cid:1) ( n + θ ) ( m − e − λ m t , n ≥ . Theorem 5.1 (Ethier and Griffiths [3], [4])
Assume that t > and µ is in M ( S ) with µ ( S ) > . Set ν ( · ) = µ ( · ) µ ( S ) . (1) The transition function of the MBI process Y t is Q ( t, µ, · ) = q µ ( S ) ,λ Γ C ( − λ,t ) θ,ν ( · ) (5.2)+ ∞ X n =1 q µ ( S ) ,λn ( t ) Z S n ( µµ ( S ) ) n ( d x × · · · × d x n )Γ C ( − λ,t ) n + θ, nθ + n η n + θθ + n ν ( · ) where η n = n P ni =1 δ x i . The probability Γ βθ,ν is the reversible measure for Y t . (2) The transition function of the FVP process X t is Q ( t, ν, · ) = d θ ( t )Π θ,ν ( · ) (5.3)+ ∞ X n =1 d θn ( t ) Z S n ν n ( d x × · · · × d x n )Π n + θ, nθ + n η n + θθ + n ν ( · ) , and Π θ,ν is the reversible measure of the process.
13t is clear from this theorem that for any t > , Y t and X t are mixtures of gammaprocesses and the Dirichlet processes respectively. A comparison between Q ( t, µ, · ) and Q ( t, ν, · ) reveals a termwise Gamma-Dirichlet structure betweenΓ C ( − λ,t ) n + θ, nθ + n η n + θθ + n ν ( · ) and Π n + θ, nθ + n η n + θθ + n ν ( · ) . Let C ([0 , ∞ ) , M ( S )) denote the space of all M ( S )-valued continuous functions on [0 , ∞ )equipped with the topology of uniform convergence on compact sets. Define ̺ ( µ ( · )) = inf { t > µ t ( S ) = 0 } , µ · ∈ C ([0 , ∞ ) , M ( S ))and C ([0 , ∞ ) , M ( S )) = { µ · ∈ C ([0 , ∞ ) , M ( S )) : Z ̺ ( µ ( · ))0 d uµ u = ∞} . Define ς ( t ) : C ([0 , ∞ ) , M ( S )) → [0 , ∞ ) , t = Z ς ( t )0 d uµ u ( S ) . It is shown in [16] that the process Y ς ( t ) Y ς ( t ) ( S ) is the FVP starting at ν = µµ ( S ) . A natural questionraised in [4] is whether one can derive (5 .
3) directly from (5 .
2) using Shiga’s normalizationand random time change. Motivated by this problem, we obtain a direct derivation of { d θn ( t ) : n = 0 , , . . . } from { q µ ( S ) ,λn ( t ) : n = 0 , , . . . } through a random time change. Thisresult is then used to link the distribution of Y t and X t at every fixed time t > N ( t ) for the time inhomogeneous Markov chain N µ ( S ) ( t ). Define τ t : [0 , ∞ ) → [0 , ∞ ) by t = Z τ t d u ( N ( u ) ∨ θ − C ( − λ, u ) . Theorem 5.2
The process N ( τ t ) is the embedded chain of Kingman’s coalescent D θt . Proof:
Let C ( ˆ Z + ) be the set of all continuous functions on ˆ Z + and C = { f ∈ C ( ˆ Z + ) : lim n →∞ n ( f ( n − − f ( n )) exists } . For any t > f in C , defineΩ t f ( m ) = ( , m = ∞ m C ( − λ,t ) ( f ( m − − f ( m )) , m ∈ Z + t is the time dependent infinitesimal generator of N ( t ). Hence for any f in C and t, s > f ( N ( t + s )) − f ( N ( s )) − Z t Ω s + u f ( N ( s + u )) d u (5.4)is a martingale with respect to the natural filtration of { N ( t + s ) : t ≥ } . Let τ t ( s ) be givenby t = Z s + τ t ( s ) s d u ( N ( u ) ∨ θ − C ( − λ, u ) . Clearly τ t ( s ) converges to τ t as s tends to zero. It follows from (5 .
4) that f ( N ( s + τ t ( s ))) − Z τ t ( s )0 Ω s + u f ( N ( s + u )) d u = f ( N ( s + τ t )) − Z t (Ω s + τ u ( s ) f ( N ( s + τ u ( s ))))( N ( s + τ u ( s )) ∨ θ − C ( − λ, s + τ u ( s )) d u = f ( N ( s + τ t ( s ))) − Z t λ N ( s + τ u ( s )) [ f ( N ( s + τ u ( s )) − − f ( N ( s + τ u ( s )))] d u is a martingale with respect to the filtration generated by { N ( τ t ( s ) + s ) : t ≥ } . Set˜Ω f ( m ) = ( λ m ( f ( m − − f ( m )) m ∈ Z + , lim n →∞ λ n [ f ( n − − f ( n )] m = ∞ It then follows by letting s tend to zero that the process N ( τ t ) is a solution to the martingaleproblem associated with the generator ˜Ω. On the other hand, based on the proof of lemma2.5 in [3], the process D θt is the unique solution of the martingale problem associated with˜Ω. Therefore the theorem holds. ✷ As an application of this result, we get the following derivation of the fixed time distri-bution of the FVP process from the MBI process.
Theorem 5.3
For any t > , let ν ( θ, t ) = N ( t ) η N ( t ) + θν N ( t ) + θ . Then we have Y t d = Y C ( − λ,t ) N ( t )+ θ,ν ( θ,t ) (5.5) X t d = Y C ( − λ,t ) N ( τ t )+ θ,ν ( θ,τ t ) Y C ( − λ,t ) N ( τ t )+ θ,ν ( θ,τ t ) ( S ) . (5.6)15 emarks In [16], the MBI process Y t can be represented as the sum of two independentPoisson clusters with one corresponding to the mass distribution of the descendants of theoriginal population and the other the mass distribution of all immigrants. The random timechange in [16] is for the whole process Y t while the random time change here is only for thePoisson number of descendants. The scaling parameter C ( − λ, t ) plays no role in (5 .
6) andtherefore the random time change involves N ( t ) only. The Fleming-Viot process is a large class of probability-valued processes that describe theevolution of a population under the influence of mutation, selection, recombination, andrandom sampling. If there is no selection and recombination, then the FVP process X t isshown in [11] to be the only reversible Fleming-Viot process. The corresponding reversiblemeasure is Π θ,ν . In [10], the reversibility of X t is shown to be equivalent to the quasi-invariance of the Dirichlet process Π θ,ν . In this subsection we study the reversibility of aclass of branching diffusions with immigration and investigate the corresponding relationbetween reversibility and quasi-invariance.For any µ in M ( S ), a ( · ) , b ( · ) in C ( S ) satisfying a ( · ) > b ( · ) >
0, consider thefollowing generator˜ L F ( µ ) = Z S µ ( d s ) a ( s ) δ F ( µ ) δµ ( s ) + Z S ( µ ( d s ) − µ ( d s ) b ( s )) δF ( µ ) δµ ( s ) (5.7)with domain D ( ˜ L ) = { φ ( h µ, f i , . . . , h µ, f n i ); µ ∈ M ( S ) , n ≥ , f i ∈ B ( S ) , φ ∈ C ( R n ) } . It is known that the martingale problem associated with ˜ L is well-posed (cf. [16]).The unique solution to the martingale problem is a diffusion process Z t with fixed timedistribution characterized by E µ [ e −h Z t ,f i ] = exp (cid:16) − D µ , a − log (cid:16) ab (1 − e − bt ) f (cid:17)E(cid:17) (5.8) × exp (cid:18) − (cid:28) µ, e − bt a/b (1 − e − bt ) f (cid:29)(cid:19) . The corresponding carr´e du champ isΓ(
F, G ) = 12 { ˜ L ( F G ) − G ˜ L F − F ˜ L G } (5.9)= (cid:28) µ, a ( s ) δF ( µ ) δµ ( s ) δG ( µ ) δµ ( s ) (cid:29) , F, G ∈ D ( ˜ L ). It follows from direct calculation that for any F, G, H in D ( ˜ L )Γ( F H, G ) + Γ(
HG, F ) − Γ( H, F G ) = 2 H Γ( F, G ) . (5.10)For any f in B ( S ) and µ in M ( S ), define S f : M ( S ) → M ( S ) , µ ( d s ) e f ( s ) µ ( d s )and Λ( µ, f ) = h µ , f /a i − h µ, ( e f − b/a i . Definition 5.1
Let Ξ be a probability on M ( S ) . The operator ˜ L is reversible with respectto Ξ if Z F ˜ L G Ξ( d µ ) = Z G ˜ L F Ξ( d µ ) , F, G ∈ D ( ˜ L ) . The probability Ξ is Λ -quasi-invariant with respect to the family of transformations { S f : f ∈ B ( S ) } if Ξ and the image law S f (Ξ) of Ξ under S f are mutually absolutely continuous withdensity given by d S f (Ξ) d Ξ ( µ ) = e Λ( µ, − f ) . The next result shows that the reversibility and the Λ-quasi-invariance are equivalent.
Theorem 5.4
A Borel probability measure Ξ on M ( S ) is reversible with respect to ˜ L if andonly if Ξ is Λ -quasi-invariant with respect to the family of transformations { S f : f ∈ B ( S ) } . Proof:
We first show that the reversibility implies the Λ-quasi-invariance. From thedefinition of the carr´e du champ, it is clear that for any
F, G in D ( ˜ L ) Z M ( S ) [Γ( F, G ) + F ˜ L G ]Ξ( d µ ) = 0 . (5.11)For any t ≥ , g ∈ B ( S ), let µ t = S − atg ( µ )and F ( µ ; f , . . . , f n ) = φ ( h µ, f i , . . . , h µ, f n i ) , where n ≥ , φ ∈ D ( ˜ L ) , f i ∈ B ( S ) , i = 1 , , . . . . It is clear that F ( µ t ; f , . . . , f n ) = F ( µ ; e − atg f , . . . , e − atg f n ) . By direct calculation 17 F ( µ t ; f , . . . , f n ) d t = − (cid:28) µ, ag δF ( µ t ; f , . . . , f n ) δµ ( s ) (cid:29) . (5.12)For G ( µ ) = h µ, g i , we have ˜ L G ( µ ) = h µ , g i − h µ, bg i (5.13)and Λ( µ t , atg ) = h µ , g i t − h µ, ba (1 − e − atg ) i (5.14)= Z t ˜ L G ( µ u ) d u. Set F t ( µ ) = F ( µ t ; f , . . . , f n ) e − Λ( µ t ,atg ) H ( t ) = Z M ( S ) F t ( µ )Ξ( d µ ) . Since d ˜ L G ( µ t ) d t = − * µ, ag δ ˜ L G ( µ t ) δµ ( s ) + , it follows from (5 .
12) and (5 .
14) that (cid:28) µ, ag δF t ( µ ) δµ ( s ) (cid:29) = (cid:28) µ, δF ( µ t ; f , . . . , f n ) δµ ( s ) (cid:29) e − Λ( µ t ,atg ) − F t ( µ ) Z t * µ, ag δ ˜ L G ( µ s ) δµ ( s ) + d s = − d F ( µ t ; f , . . . , f n ) d t e − Λ( µ t ,atg ) + F t ( µ )( ˜ L G ( µ t ) − ˜ L G ( µ )) . Consequently, H ′ ( t ) = Z M ( S ) (cid:20) ddt F ( µ t ; f , . . . , f n ) e − Λ( µ t ,atg ) − F t ( µ ) ˜ L G ( µ t ) (cid:21) Ξ( dµ )= Z M ( S ) (cid:20) −h µ, ag δF t ( µ ) δµ ( s ) i − F t ( µ ) ˜ L G ( µ ) (cid:21) Ξ( dµ )= − Z M ( S ) h Γ( F t , G ) + F t ( µ ) ˜ L G ( µ ) i Ξ( dµ )Since F t , G ∈ D ( L ), it follows from (5 .
11) that H ′ ( t ) = 0. In particular we have Z F ( µ ; f , . . . , f n ) e − Λ( µ ,ag ) Ξ( dµ ) = Z F ( µ ; f , . . . , f n )Ξ( dµ ) . g is arbitrary and a ( x ) >
0, we have for any f ∈ B ( S ) Z F ( S − f ( µ ); f , . . . , f n ) e − Λ( S − f µ,f ) Ξ( dµ ) = Z F ( µ ; f , . . . , f n )Ξ( dµ ) , Replacing F ( µ ; f , . . . , f n ) by F ( µ ; f , . . . , f n ) e Λ( µ,f ) , we obtain Z M ( S ) F ( S − f µ ; f , . . . , f n )Ξ( dµ ) = Z F ( µ ; f , . . . , f n ) e Λ( µ,f ) Ξ( dµ ) , which shows that Ξ is Λ-quasi-invariant.Next we show that the Λ-quasi-invariance implies reversibility. Assume that Ξ is Λ-quasi-invariant. Then we have H ′ ( t ) = 0 for all t ≥
0. In particular, we have for any g in B ( S )and F in D ( ˜ L ) − H ′ (0) = Z M ( S ) [ h µ, ag δF ( µ ) δµ ( x ) i + F ( µ ) ˜ L G ( µ )]Ξ( dµ )= Z M ( S ) [Γ( F, G ) + F ( µ ) ˜ L G ( µ )]Ξ( dµ ) = 0 , where as before G ( µ ) = h µ, g i . For any g , g in B ( S ), let G i ( µ ) = h µ, g i i , i = 1 , . Then itfollows from (5 .
10) that Z M ( S ) F ˜ L ( G G )Ξ( dµ ) = Z M ( S ) F (2Γ( G , G ) + G ˜ L G + G ˜ L G )Ξ( dµ )= Z M ( S ) (Γ( F G , G ) + Γ( F G , G ) − Γ( F, G G ))Ξ( dµ ) − Z M ( S ) Γ( F G , G ) + Γ( F G , G )Ξ( dµ )= − Z Γ( F, G G )Ξ( dµ ) . By induction, Z M ( S ) F ˜ L G Ξ( d µ ) = − Z M ( S ) Γ( F, G )Ξ( d µ ) (5.15)for G of the form Q ni =1 h µ, g i i , g i ∈ B ( S ) , n ≥
1. Note that every function φ in C ( R n ) can beapproximated by polynomials under the Sobolev norm of order two, so the equality (5 . G in D ( ˜ L ). Consequently, Ξ is reversible with respect to ˜ L . ✷ As an application of Theorem 5.4, we consider the reversibility of the process Z t . Letting t tend to infinity in (5 . Z t has a unique invariant distributionΓ a − b,a − µ characterized by 19 Γ a − b,a − µ [ e −h µ,h i ] = exp (cid:16) − D µ , a − log(1 + ab h ) E(cid:17) , h ∈ B + ( S ) . (5.16)Next we establish the reversibility of Z t by verifying the Λ-quasi-invariance of Γ a − b,a − µ . Theorem 5.5
The law Γ a − b,a − µ is Λ -quasi-invariant and therefore the process Z t is re-versible with respect to Γ a − b,a − µ . Proof:
For any f in B ( S ) and h in B + ( S ), it follows from (5 .
16) that E S f (Γ a − b,a − µ ) [ e −h µ,h i ]= E Γ a − b,a − µ [ e −h S f ( µ ) ,h i ]= E Γ a − b,a − µ [ e −h µ,e f h i ]= exp {− D µ , a − log(1 + ab he f ) E } = exp {−h µ , a − f i} exp (cid:18) − (cid:28) µ , a − log (cid:18) ab ( h + ba ( e − f − (cid:19)(cid:29)(cid:19) = exp {−h µ , a − f i} E Γ a − b,a − µ [ e −h µ, ( h + ba ( e − f − i ]= E Γ a − b,a − µ [ e Λ( µ, − f ) e −h µ,h i ] , which yields the result. ✷ Remarks
The Λ-quasi-invariance is established in [19] when a and b are constants. For theMBI process Y t , we have a ( s ) = , b ( s ) = λ , and µ = ν .We conclude this subsection with a derivation of the reversibility of the FVP processfrom the reversibility of the MBI process exploiting the Gamma-Dirichlet structure. Theorem 5.6
The reversibility of the MBI process implies the reversibility of the FVP pro-cess.
Proof:
First recall that the domain of the generator A for the FVP process is given by D ( A ) = { φ ( h ν, f i , . . . , h ν, f n i ); ν ∈ M ( S ) , n ≥ , f i ∈ B ( S ) , φ ∈ C ( R n ) } . For any µ ∈ M ( S ) \ { } and any Φ , Ψ in D ( A ), define r ( µ ) = h µ, i F ( µ ) = r ( µ )Φ( µµ ( S ) ) G ( µ ) = r ( µ )Ψ( µµ ( S ) ) , F (0) = G (0) = 0. Then it is clear that both F and G belong to D ( L ). By the definitionof D ( A ), there are n, m ≥ , f , . . . , f n , g , . . . , g m ∈ B ( S ) , φ, ψ ∈ C ( R n ) such thatΦ( ν ) = φ ( h ν, f i , . . . , h ν, f n i )Ψ( ν ) = ψ ( h ν, g i , . . . , h ν, g m i ) . Fix a µ in M ( S ) \ { } . As in section 2.2, we write ˆ µ = µµ ( S ) . Set˜Φ( µ ) = Φ(ˆ µ ) , ˜Ψ( µ ) = Ψ(ˆ µ ) . Then by direct calculation δF ( µ ) δµ ( s ) = 3 r ( µ ) ˜Φ( µ ) + r ( µ ) δ ˜Φ( µ ) δµ ( x )= 3 r ( µ ) ˜Φ( µ ) + r ( µ ) n X i =1 ∂ i φ ( f i − h ˆ µ, f i i )and δ F ( µ ) δµ ( s ) = 6 r ( µ ) ˜Φ( µ ) + 4 r ( µ ) n X i =1 ∂ i φ ( f i − h ˆ µ, f i i )+ r ( µ ) n X i,j =1 ∂ ij φ ( f i − h ˆ µ, f i i )( f j − h ˆ µ, f j i ) . Substituting this into (1 . L F ( µ ) = r ( µ ) A Φ(ˆ µ ) + 3 r ( µ )( θ − λr ( µ )2 + 2)Φ(ˆ µ ) . (5.17)Similarly L G ( µ ) = r ( µ ) A Ψ(ˆ µ ) + 3 r ( µ )( θ − λr ( µ )2 + 2)Ψ(ˆ µ ) . (5.18)For µ = 0, we have L F ( µ ) = L G ( µ ) = 0. By the reversibility of the MBI process0 = Z M ( S ) [ G ( µ ) L F ( µ ) − F ( µ ) L G ( µ )]Γ βθ,ν ( d µ )= Z M ( S ) r ( µ )[Ψ(ˆ µ ) A Φ(ˆ µ ) − Φ(ˆ µ ) A Ψ(ˆ µ )]Γ βθ,ν ( d µ ) , Z M ( S ) r ( µ )[Ψ(ˆ µ ) A Φ(ˆ µ ) − Φ(ˆ µ ) A Ψ(ˆ µ )]Γ βθ,ν ( d µ )= Z ∞ θ ) β θ x θ +4 e − x/β d x Z M ( S ) [Ψ(ˆ µ ) A Φ(ˆ µ ) − Φ(ˆ µ ) A Ψ(ˆ µ )]Π θ,ν ( d ˆ µ )= β Γ(5 + θ )Γ( θ ) Z M ( S ) [Ψ(ˆ µ ) A Φ(ˆ µ ) − Φ(ˆ µ ) A Ψ(ˆ µ )]Π θ,ν ( d ˆ µ ) = 0 . Therefore the FVP process is reversible with respect to Π θ,ν . ✷ Remark
It is not clear whether the reversibility of the MBI process follows from the re-versibility of the FVP process.
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