Additional material on local limit theorem for finite Additive Markov Processes [HL13]

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Additional material on local limit theorem for finite Additive Markov Processes [HL13]

Loïc Hervé, James Ledoux

To cite this version:

Loïc Hervé, James Ledoux. Additional material on local limit theorem for finite Additive Markov Processes [HL13]. 2013. �hal-00825592v2�

Additional material on local limit theorem for finite Additive Markov Processes [HL13]

Loïc HERVÉ and James LEDOUX^∗ June 22, 2013

Abstract

This paper proposes additional material to the main statements of [HL13] which are are recalled in Section 2. In particular an application of [HL13, Th 2.2] to Renewal Markov Processes is provided in Section 4 and a detailed checking of the assumptions of [HL13, Th 2.2] for the joint distribution of local times of a finite jump process is reported in Section 5. A uniform version of [HL13, Th 2.2] with respect to a compact set of transition matrices is given in Section 6 (see [HL13, Remark 2.4]). The basic material on the semigroup of Fourier matrices and the spectral approach used in [HL13] is recalled in Section 3 in order to obtain a good understanding of the properties involved in this uniform version.

Keywords: Gaussian approximation, Local time, Spectral method, Markov random walk.

Contents

1 Notations 2

2 The LLT for the density process 2

3 Semigroup of Fourier matrices and basic lemmas 4

4 Application to the Markov Renewal Processes 7

5 Application to the local times of a jump process 9

5.1 The local limit theorem for the density of local times . . . . 9

5.2 The main lines of the derivation of Proposition 5.1 . . . . 11

5.2.1 Checking Condition (AC2) . . . . 12

5.2.2 Checking Condition (AC1) . . . . 13

6 A uniform LLT with respect to transition matrix P 15

A Proof of Lemma 5.2 18

B Proof of Lemma 5.3 19

C Linear transformation of MAPs 22

∗Université Européenne de Bretagne, I.R.M.A.R. (UMR-CNRS 6625), Institut National des Sciences Ap- pliquées de Rennes. Loic.Herve,[email protected]

1 Notations

Any vectorv= (v_k)k∈{1,...,N} ∈C^N is a considered as a row-vector andv^⊤is the corresponding column-vector. The vector with all components equal to 1 is denoted by 1. The euclidean product scalar and its associated norm onC^N is denoted by h·,·i and k · k respectively. The set of N ×N-matrices with complex entries is denoted by MN(C). Let k · k∞ denote the supremum norm on C^N: ∀v ∈ C^N, kvk∞ = maxk∈{1,...,N}|v_k)|. For the sake of simplicity, k · k∞ also stands for the associated matrix norm:

∀A∈ MN(C), kAk∞ := sup

kvk∞=1kAv^⊤k∞. We also use the following norm k · k0 onMN(C):

∀A= (A_k,ℓ)(k,ℓ)∈{1,...,N}² ∈ MN(C), kAk0 := max

(k,ℓ)∈{1,...,N}|A_k,ℓ|. It is easily seen that

∀A∈ MN(C), kAk0 ≤ kAk∞≤NkAk0. (1) For any bounded positive measure ν onR^d, we define its Fourier transform as:

∀ζ ∈R^d, bν(ζ) :=

Z

R^d

e^ihζ,yidν(y).

Let A = (Ak,ℓ)(k,ℓ)∈{1,...,N}² be a N ×N-matrix with entries in the set of bounded positive measures onR^d. We set

∀B ∈B(R^d), A(1_B) := Ak,ℓ(1_B)

(k,ℓ)∈X2 (2a)

∀ζ ∈R^d, Ab(ζ) := (Abk,ℓ(ζ))_(k,ℓ)∈X2. (2b) We denote by ⌊t⌋ the integer part of anyt∈T.

2 The LLT for the density process

Let{(X_t, Z_t)}t∈Tbe an MAP with state spaceX×R^d, whereX:={1, . . . , N}and the driving Markov process {X_t}t∈T has transition semi-group {P_t}t∈T. We refer to [Asm03, Chap. XI]

for the basic material on such MAPs. The conditional probability to {X0 = k} and its associated expectation are denoted by P_k and E_k respectively. Note that if T :R^d→R^m is a linear transformation, then {X_t, T(Z_t)}t∈T is still a MAP on X×R^m (see Lemma C.1).

We suppose that {X_t}t∈T has a unique invariant probability measure π. Set m :=E_π[Z₁] = P

kπ(k)E_k[Z₁]∈ R^d. Consider the centered MAP{(X_t, Y_t)}t∈T where Y_t := Z_t−t m. The two next assumptions are involved in both CLT and LLT.

(I-A):The stochastic N ×N-matrix P :=P₁ is irreducible and aperiodic.

(Mα): The family of r.v. {Y_v}v∈(0,1]∩T satisfies the uniform moment condition of order α:

M_α := max

k∈X sup

v∈(0,1]∩T

E_k kY_vk^α

<∞. (3)

The next theorem provides a CLT for t^−1/2Y_t, proved for d:= 1in [KW64] for T=Nand in [FH67] forT= [0,∞) (see [FHL12] forρ-mixing driving Markov processes).

Theorem 2.1 Under Assumptions (I-A) and (M2), {t^−1/2Y_t}t∈T converges in distribution to a d-dimensional Gaussian lawN(0,Σ) when t→+∞.

Now let us specify the notations and assumptions involved in our LLT. First we assume that the MAP{(X_t, Y_t)}t∈T satisfies the following usual non-lattice condition:

(NL) : there is no a ∈ R^d, no closed subgroup H in R^d, H 6= R^d, and finally no function β : X→R^d such that:

∀k∈X, Y₁+β(X₁)−β(k) ∈ a+H P_k-a.s.,

Second we introduce the assumptions on the density process of t^−1/2Yt. For any t ∈ T and (k, ℓ)∈X², we define the bounded positive measure Yk,ℓ,t onR^d:

∀B∈B(R^d), Yk,ℓ,t(1B) :=P_k

Xt=ℓ, Yt∈B . (4)

Let ℓ_d denote the Lebesgue measure on R^d. From the Lebesgue decomposition of Yk,ℓ,t

w.r.t.ℓ_d, there are two bounded positive measures Gk,ℓ,t and µ_k,ℓ,t on R^dsuch that

∀B ∈B(R^d) Yk,ℓ,t(1B) :=Gk,ℓ,t(1B) +µ_k,ℓ,t(1B) (5a) where Gk,ℓ,t(1_B) =

Z

B

g_k,ℓ,t(y)dy (5b)

for some measurable function g_k,ℓ,t :R^d→[0,+∞), and such that µ_k,ℓ,t and ℓ_d are mutually singular. The measure Gk,ℓ,t is called the absolutely continuous (a.c.) part of Yk,ℓ,t with associated density g_k,ℓ,t. For any t ∈ T, we introduce the following N ×N-matrices with entries in the set of bounded positive measures on R^d

Yt:= (Yk,ℓ,t)_(k,ℓ)∈X2, Gt:= (Gk,ℓ,t)_(k,ℓ)∈X2, Mt:= (µ_k,ℓ,t)_(k,ℓ)∈X2, (6a) and for everyy ∈R^d, the following realN ×N-matrix:

G_t(y) := g_k,ℓ,t(y)

(k,ℓ)∈X2. (6b)

Then the component-wise equalities (5a)-(5b) read as follows in a matrix form: for anyt∈T

∀B ∈B(R^d), Yt(1_B) =Gt(1_B) +Mt(1_B) = Z

B

G_t(y)dy+Mt(1_B). (6c) The assumptions on the a.c. partGtand the singular partMtofYtare the following ones.

AC 1 : There exist c >0 and ρ∈(0,1) such that

∀t >0, kMt(1Rd)k0 ≤cρ^t (7) and there exists t₀ >0 such that

ρ^t0max(2, cN)≤1/4 (8a)

Γt0(ζ) := sup

w∈[t0,2t0)kGbw(ζ)k0 −→0 whenkζk →+∞. (8b)

AC 2 : For any t > 0, there exists an open convex subset Dt of R^d such that G_t vanishes on R^d\ Dt, where Dt denotes the adherence of Dt. Moreover G_t is continuous on Dt and differentiable on Dt, with in addition

sup

t>0

sup

y∈Dt

kG_t(y)k0 <∞ (9a) sup

y∈∂Dt

kG_t(y)k0 =O t^−(d+1)/2

where ∂Dt:=Dt\ Dt (9b) j= 1, . . . , d: sup

t>0

sup

y∈Dt

∂G_t

∂yj

(y)

0<∞. (9c)

The next theorem gives a LLT for the density of the a.c. part of the probability distribution oft^−1/2Y_t. This is the main contribution of [HL13].

Theorem 2.2 Let {(X_t, Y_t)}t∈T be a centered MAP satisfying Assumptions (I-A), (M3), (AC1)-(AC2). Moreover assume that the matrix Σ of Theorem 2.1 is invertible. Then, for every k∈X, the densityf_k,t(·) of the a.c. part of the probability distribution of t^−1/2Y_t under P_k satisfies the following property:

sup

y∈R^d

f_k,t(y)−η_Σ(y)≤O t^−1/2

+O sup

y /∈Dt

η_Σ(t^−1/2y)

(10) where η_Σ(·) denotes the density of the non-degenerate d-dimensional Gaussian distribution N(0,Σ) involved in Theorem 2.1.

3 Semigroup of Fourier matrices and basic lemmas

We assume that the conditions of Theorem 2.2 hold, and for the sake of simplicity that Σis the identity matrix. The proof of Theorem 2.2 involves Fourier analysis as in the i.d.d. case.

In our Markov context, this study is based on the semi-group property of the matrix family {Ybt(ζ)}t∈T for every ζ ∈ R^d which allows to analyze the characteristic function of Y_t. In this section, we provide a collection of lemmas which highlights the connections between the assumptions of Section 2 and the behavior of this semi-group. This is the basic material for the derivation of Theorem 2.2 in [HL13, Section 3].

Recall that, for every (k, ℓ)∈X², the measuresGk,ℓ,t (with densityg_k,ℓ,t) andµ_k,ℓ,t denote the a.c. and singular parts of the bounded positive measureYk,ℓ,t(1_·) =P_k{X_t=ℓ, Y_t∈ ·}on R^d. Using the notation of (5a)-(5b), the bounded positive measureP_N

ℓ=1Gk,ℓ,t with density g_k,t:=

XN

ℓ=1

g_k,ℓ,t

is the a.c. part of the probability distribution of Y_t under P_k, while µ_k,t :=PN

ℓ=1µ_k,ℓ,t is its singular part. That is, we have for any k∈Xand t >0:

∀B ∈B(R^d), P_k{Y_t∈B}= Z

B

g_k,t(y)dy+µ_k,t(1_B). (11) The bounded positive measure Yt is defined in (6a) and its Fourier transform Ybt is (see (2b)):

∀(t, ζ)∈T×R^d, ∀(k, ℓ)∈X², Ybt(ζ)

k,ℓ=E_k

1_{Xt=ℓ}e^ihζ,Yti

. (12)

Note that Ybt(0) = P_t. From the additivity of the second component Y_t, we know that that {Ybt(ζ)}t∈T,ζ∈R^d is a semi-group of matrices (e.g. see [FHL12] for details), that is using the usual product in the spaceMN(C) of complexN ×N-matrices:

∀ζ ∈R^d, ∀(s, t)∈T², Ybt+s(ζ) =Ybt(ζ)Ybs(ζ). (SG) In particular the following property holds true

∀ζ ∈R^d, ∀n∈N, Ybn(ζ) := E_k[1_{Xn=ℓ}e^ihζ,Yni]

(k,ℓ)∈X2 =Yb1(ζ)ⁿ. (13) For everyk∈Xand t∈T, we denote by φ_k,t the characteristic function ofYt under P_k:

∀ζ ∈R^d, φ_k,t(ζ) :=E_k

e^ihζ,Yti

=e_kYbt(ζ)1^⊤, (14) where e_k is thek-th vector of the canonical basis of R^d.

The first lemma below provides the control ofφ_k,t on a ball B(0, δ) :={ζ ∈R^d:kζk< δ} for some δ > 0, using a perturbation approach. The second one is on the control of φ_k,t on the annulus {ζ ∈ R^d :δ ≤ kζk ≤ A] for any A > δ. Finally the third lemma focuses on the Fourier transform of the density g_k,ℓ,t of the a.c. part Gk,ℓ,t of Yk,ℓ,t in (5a) on the domain {ζ ∈R^d:kζk ≥A} for someA >0.

Lemma 3.1 ([HL13, Lem. 4.1]) Under Assumptions (I-A) and(M3), there exists a real number δ >0such that, for all ζ ∈B(0, δ), the characteristic function of Y_t satisfies

∀k∈X, ∀t∈T, φ_k,t(ζ) =λ(ζ)^⌊t⌋L_k,t(ζ) +R_k,t(ζ), (15) with C-valued functions λ(·), L_k,t(·) and R_k,t(·) on B(0, δ) satisfying the next properties for t∈T,ζ ∈R^d andk∈X:

kζk< δ ⇒ λ(ζ) = 1−kζk²

2 +O(kζk³) (16a)

t≥2, kt^−1/2ζk< δ ⇒ λ(t^−1/2ζ)^⌊t⌋−e^−kζk2/2≤C (1 +kζk³)e^−kζk2/8

√t (16b)

kζk< δ ⇒ L_k,t(ζ)−1 ≤Ckζk (16c)

∃r∈(0,1), sup

kζk≤δ |R_k,t(ζ)| ≤C r^⌊t⌋ (16d)

where the constantC >0 in (16b)-(16d) only depends on δ and onM₃ in (M3) (see (3)).

Proof of (16b). From (16a) we obtain for v∈R^d such that |v| ≤δ (up to reduce δ)

|λ(v)| ≤1−kvk²

2 +kvk²

4 ≤e^−kvk2/4.

Therefore we have for any t≥2 and anyζ ∈R^dsuch that t^−1/2kζk ≤δ, λ t^−1/2ζ≤e^−kζk

2

4t . (17)

Now consider anyt≥2and any ζ ∈R^d such that t^−1/2kζk ≤δ. Set n:=⌊t⌋, and write λ t^−1/2ζn

−e^−kζk

2

2 = λ t^−1/2ζ

−e^−kζk

2 2n ⁿ⁻¹X

k=0

λ t^−1/2ζn−k−1

e^−kkζk

2 2n .

We have

ⁿ⁻¹X

k=0

λ t^−1/2ζn−k−1

e^−kkζk

2

2n ≤

n−1X

k=0

λ t^−1/2ζ^n−k−1e^−kkζk

2 2n

≤

n−1X

k=0

e⁻kζk2(n−k−1) 4t e^−kkζk

2 4t

≤ n e^−nkζk

2 4t e^kζk

2

4t ≤b t e^−kζk

2 8

where b := sup_|u|≤δexp(u²/4), since n/t≥(t−1)/t≥1/2 for t ≥2. Moreover, using (16a) and the fact that|e^a−e^b| ≤ |a−b|for anya, b∈(−∞,0), we obtain that there exist positive constants Dand D^′ such that

λ t^−1/2ζ

−e^−kζk

2

2n ≤ λ t^−1/2ζ

−e^−kζk

2

2t +e^−kζk2t2 −e^−kζk

2 2n

≤ D t⁻²³kζk³+kζk² 2

1 t − 1

n

≤ D t⁻²³kζk³+kζk² 2

1 t(t−1)

≤ D^′ t⁻³²kζk³+t⁻²kζk² . Thus

λ t^−1/2ζn

−e^−kζk

2 2

≤ bD^′t e^−kζk

2

8 t⁻³²kζk³+t⁻²kζk²

≤ bD^′e^−kζk

2 8

kζk³

√t +kζk² t

≤ bD^′

√t e^−kζk

2

8 kζk³+kζk²

≤ D^′′

√t e^−kζk

2

8 1 +kζk³

for some positive constantD^′′.

Lemma 3.2 ([HL13, Lem. 4.2]) Assume that Conditions (I-A), (AC1) and (Mα) for some α > 0 hold. Let δ, A be any real numbers such that 0 < δ < A. There exist constants D≡D(δ, A) >0 andτ ≡τ(δ, A)∈(0,1) such that

∀k∈X, ∀t∈T, sup

δ≤kζk≤A|φ_k,t(ζ)| ≤D τ^⌊t⌋. (18)

Lemma 3.3 ([HL13, Th. 4.3]) Under Condition (AC1), there exist positive constants A andC such that the following property holds:

|ζ| ≥A =⇒ ∀(k, ℓ)∈X², ∀t∈[t₀,+∞[,

bg_k,ℓ,t(ζ)≤C t 2^t/t0.

4 Application to the Markov Renewal Processes

Let{X_n, Y_n}n∈Nbe a discrete-time MAP with state spaceX×RandX:={1, . . . , N}. When {ξ_n}n≥0, with ξ₀ :=Y₀ and ξ_n:=Y_n−Y_n−1 for n≥1, is a sequence of non-negative random variables, then{X_n, Y_n}n∈Nis also known as a Markov Renewal Process (MRP). The so-called semi-Markov kernelQ(·;{·} ×dy) is defined by (e.g. see [Asm03, VII.4])

∀B ∈B(R), ∀(k, ℓ)∈X², P_k{X₁ =ℓ, Y₁∈B}= Z

B

Q(k;{ℓ} ×dy) (19) and is nothing else but the measure Yk,ℓ,1 in (4). The transition probability matrix P asso- ciated with the Markov chain{X_n}n∈N is given by

∀(k, ℓ)∈X², P(k, ℓ) =Q(k;{ℓ} ×R) = Z

R

Q(k;{ℓ} ×dy).

For any n≥2, the bounded positive measure Yk,ℓ,n is defined by the convolution product of the semi-Markov kernelQ, that is

∀B ∈B(R^d), Yk,ℓ,n(1_B) :=P_k{X_n=ℓ, Y_n∈B}=Q^⋆n(k,{ℓ} ×B) (20) Then, the Theorem 2.2 for the density process ofYn/√

ncould be specified to this specific class of MAPs. Since we only have to replace timetbynin all the material developed in Section 2, we omit the details. Note that the only simplification in assumptions of Theorem 2.2 is on the uniform moment condition(Mα)which reduces to: the r.v.Y1 satisfies the following moment condition of orderα:

M_α:= max

k∈XE_k kY₁k^α

<∞. (21)

We will only illustrate our main result on the MRP embedded in a Markovian Arrival Process (e.g. see [Asm03, XI.1]).

Recall that a N-state Markovian Arrival Process is a continuous-time MAP{(J_t, N_t)}t≥0

on the state space{1, . . . , N} ×N, where N_t represents the number of arrivals up to time t, while the states of the driving Markov process{Jt}t≥0 are called phases. Let Yn be the time at the nth arrival (Y₀ = 0 a.s.) and let X_n be the state of the driving process just after the nth arrival. Then {(X_n, Y_n)}n∈N is known to be an MRP with the following semi-Markov kernel Qon {1, . . . , N} ×R: for any (k, ℓ)∈X²

Q(k;{ℓ} ×dy) := (e^yD0D₁)(k, ℓ) 1_(0,∞)(y)dy =e_ke^yD0D₁e_ℓ^⊤1_(0,∞)(y)dy (22) which is parametrized by a pair of N ×N-matrices usually denoted by D₀ and D₁. Such a process is also an instance of Markov Random Walk. The matrixD₀+D₁ is the infinitesimal generator of the background Markov process {J_t}t≥0. Matrix D₀ is always assumed to be stable andD₀+D₁ to be irreducible. In this case,{J_t}t≥0 has a unique invariant probability measure denoted by π. The process {X_n}n∈N is a Markov chain with state space X :=

{1, . . . , N}and transition probability matrix P:

∀(k, ℓ)∈X², P(k, ℓ) =Q(k;{ℓ} ×R) = (−D₀)⁻¹D₁

(k, ℓ). (23)

This Markov chain has an invariant probability measure φ (different of π). From (22), the bounded positive measureYk,ℓ,1is absolutely continuous with respect to the Lebesgue measure

on Rwith densityg_k,ℓ,1(y) =e_ke^yD0D₁e_ℓ^⊤1_(0,∞)(y). Then matrix G_t defined in (6b) has the form

G1(y) =e^yD0D11_(0,∞)(y).

For any n ≥ 2, the positive measure Yk,ℓ,n in (20) is absolutely continuous with respect to the Lebesgue measure with density given by

∀y ∈R, G_n(y) = (G_n−1⋆ G₁)(y) =: X

j∈X

G_k,j,1⋆ G_j,ℓ,n−1 (y)

k,ℓ∈X2

. (24)

Let us check the assumptions of Theorem 2.2. Assumption(I-A)onP in (23) is standard in the literature on the Markov Arrival Process. It is well known that Y₁ has a moment of order 3given by E_k[(Y₁)³] = 3!e_k(−D₀)⁻³1^⊤, so that Condition (21) holds with α= 3. Let us give some details for checking Conditions (AC1)-(AC2).

(AC2): The open convex Dn involved in (AC2)is given for any n≥1 by Dn :=]0,∞[, withDn= [0,∞), ∂Dn={0}.

It is clear thatG₁(y) is continuous onD1 and differentiable on D1 with differential

∀y >0, dG₁

dy (y) =D₀e^yD0D₁. (25)

Since G_n is obtained from convolution product of G₁, the same properties are also valid for Gn. Next, we prove by induction that

sup

n>0

sup

y∈[0,+∞)kG_n(y)k0≤NkD₁k0. (26)

Forn:= 1, we have from norm equivalence (1) and ke^D0k∞≤1 that

∀y∈[0,+∞), kG₁(y)k0 = ke^yD0D₁k0 ≤ ke^yD0D₁k∞≤ ke^yD0k∞kD₁k∞

≤ NkD1k0.

Using the definition (24) of G_n(y) and the induction hypothesis, we have

|G_k,ℓ,n(y)| = X

j

Z

R|G_k,j,1(u)||G_j,ℓ,n−1(t−u)|du≤NkD1k0

Z

R

G_k,j,1(u)du

≤ NkD₁k0P_k{X₁=ℓ} ≤NkD₁k0. The proof of estimate (26) is complete.

Second, we easily obtain from (25) that sup

y∈(0,+∞)kdG₁

dy (y)k0≤ Nmax(kD₀k0,kD₁k0)2

Next, using an induction, we can obtain from the following well known property of the con- volution product

dGn

dy (y) =

G₁⋆ dGn−1

dy

(y)

that

sup

n>0

sup

y∈(0,+∞)kdG_n

dy (y)k0 ≤ Nmax(kD₀k0,kD₁k0)2

. Finally,G₁(0) =D₁ and G_n(0) = 0 for anyn≥2.

(AC1): First, observe thatµ_k,ℓ,n is 0 for any(k, ℓ)∈X² and n≥1. Thus (7) is satisfied for ρ= 0. In Condition (AC1), we only have to check that

kGb_n0(ζ)k0 −→0 when|ζ| →+∞

for n0 large enough. It follows from the convolution definition (24) ofGn that

∀n∈N, ∀ζ ∈R, Gb_n(ζ) := E_k

1_{Xn=ℓ}e^ihζ,Yni

(k,ℓ)∈X2 =Gb₁(ζ)ⁿ where

Gb1(ζ) = Z +∞

0

e^iζye^D0ydy D1

is integrable. In fact, using an integration by parts, we obtain that

∀ζ 6= 0, Gb₁(ζ) = −1 iζ

D₁+ Z +∞

0

e^iζyD₀e^yD0D₁dy so that

∀ζ6= 0, kGb₁(ζ)k0 ≤ 2kD₁k0

|ζ| .

Therefore, we deduce from norm equivalence (1) that for any integer n₀≥1, kGbn0(ζ)k0≤ (2NkD1k0)ⁿ⁰

|ζ|ⁿ⁰ −→0 when|ζ| →+∞.

5 Application to the local times of a jump process

5.1 The local limit theorem for the density of local times

Let{X_t}t≥0 be a Markov jump process with finite state space X:={1, . . . , N}and generator G. Its transition semi-group is given by

∀t≥0, P_t:=e^tG.

The local time L_t(i) at time tassociated with state i∈X, or the sojourn time in state i on the interval [0, t], is defined by

∀t≥0, L_t(i) :=

Z t 0

1_{Xs=i}ds.

It is well known that L_t(i) is an additive functional of {X_t}t≥0 and that {(X_t, L_t(i))}t≥0 is an MAP. In this section, we consider the MAP {(X_t, L_t)}t≥0 where L_t is the random vector of the local times

L_t:= (L_t(1), . . . , L_t(N)).

Note that, for allt >0, we have hL_t,1i=t, that is L_t isSt-valued where St:=

y∈[0,+∞)^N :hy,1i=t .

Let us assume that {X_t}t≥0 has an invariant probability measure π. This happens when the generator G is irreducible. Set m = (m₁, . . . , m_N) := E_π[L₁]. We define the S⁽⁰⁾t -valued centered r.v.

Y_t=L_t−tm

where S⁽⁰⁾t :=T_−tm(St) with the translation T_−tm by vector −tminR^N. Note that S⁽⁰⁾t is a subset of the hyperplane H of R^N defined by

H :=

y∈R^N : hy,1i= 0 . (27)

Let Λ be the bijective map from H into R^N−1 defined by Λ(y) := (y₁, . . . , y_N−1) for y :=

(y₁, . . . , y_N)∈H. We introduce the following (N −1)-dimensional random vector Y_t^′ := Λ(Y_t) = (Y_t(1), . . . , Y_t(N −1)).

The following lemma follows from Lemma C.1.

Lemma 5.1 The process {(X_t, Y_t^′)}t≥0 is a X× Dt-valued MAP whereDt is the open convex of R^N−1 defined by

Dt:=

y^′∈R^N−1:j= 1, . . . , N −1, y^′_j ∈(−m_jt,(1−m_j)t), hy^′,1i< m_Nt . (28) Proposition 5.1 If G and the sub-generators G_i^c_i^c := (G(k, ℓ))_k,ℓ∈{i}^c, i = 1, . . . , N are irreducible then the MAP {(X_t, Y_t^′)}t≥0 satisfies the conditions (M3) and (AC1)-(AC2).

Note that the matrixΣin the CLT for{t^−1/2Y_t^′}t>0is invertible from [HL13, Remark 2.3] and thatO(sup_{y /∈Dt}η_Σ(t^−1/2y)) =O(t^−1/2) from (28). Under the assumptions of Proposition 5.1 on the generator of{X_t}t≥0, Theorem 2.2 gives that

sup

y^′∈RN−1

f_k,t^′ (y^′)−(2π)^−(N−1)/2(det Σ)^−1/2e^{−12hy′⊤,Σy′⊤i}=O t^−1/2 ,

wheref_k,t^′ is the density of the a.c. part of the probability distribution oft^−1/2Y_t^′. An explicit form of Σis provided in [HL13, Remark 3.1].

In order to obtain a statement in terms ofY_t, we can use the following lemma. Recall that ℓ_N−1 denotes the Lebesgue measure onR^N−1 andB(H)stands for the Borelianσ-algebra on H. Let ν be the measure defined on (H, B(H)) as the image measure of ℓN−1 under Λ⁻¹, that is: for any positive andB(H)-measurable functionψ:H→R,

Z

H

ψ(y)ν(dy) :=

Z

RN−1

ψ(Λ⁻¹y^′)dy^′.

Using the bijectionΛ, the following lemma can be easily proved (see Appendix A).

Lemma 5.2 For any (k, ℓ) ∈X², the a.c. and singular parts of the Lebesgue decomposition of the measure P_k{X_t =ℓ, Y_t ∈ ·} on H with respect to ν are obtained from image measure under Λ of the a.c. and singular parts of the Lebesgue decomposition of the measure P_k{X_t= ℓ, Y_t^′ ∈ ·} on R^N−1 with respect to ℓN−1 .

Therefore we know that the density, sayf_k,t, of the a.c. part of the probability distribution of t^−1/2Y_t=t^−1/2(L_t−tm)

with respect to the measure ν on the hyperplane H is given by

∀h∈H, f_k,t(h) :=f_k,t^′ (Λh).

Finally, we have obtained the following local limit theorem forf_k,t.

Proposition 5.2 ([HL13, Prop. 3.1]) IfGand the sub-generatorsG_i^c_i^c := (G(k, ℓ))_k,ℓ∈{i}^c, i= 1, . . . , N are irreducible, then the density f_k,t of the a.c. part of the probability distribution of t^−1/2(L_t−tm) under P_k satisfies

sup

h∈H

f_k,t(h)−(2π)^−(N−1)/2(det Σ)^−1/2e^{−12hΛh⊤,ΣΛh⊤i} =O t^−1/2 .

It remains to prove that Proposition 5.1 holds true.

5.2 The main lines of the derivation of Proposition 5.1

Let us introduce the randomized Markov chain {Z_n}n∈N with state space X and transition matrix

Pe:=I+G/a witha >max(|G(j, j)|, j ∈X).

Since G is assumed to be irreducible, the transition matrix Pe is irreducible and aperiodic.

Moreover, since P(i, i)e >0 for any i∈X, the sub-stochastic matrix Pe_i^c_i^c := (Pe(k, ℓ))_k,ℓ∈{i}^c is aperiodic.

Let us define the following convex open set of R^N−1

∀t >0, Ct:=

y∈]0, t[^N−1,hy,1i< t . For any t > 0, the adherence of Ct is denoted by Ct and is Ct :=

y ∈ [0, t]^N−1,hy,1i ≤ t and the boundary ofCt is∂Ct:=Ct\Ctgiven by

∂Ct= [N i=1

y = (y₁, . . . , y_N−1)∈ Ct|y_i := 0 ∪

y∈ Ct| hy,1i =t . (29) The joint conditional distribution of (X_t, L^′_t) := (X_t, L_t(1), . . . , L_t(N −1))under P_k is 0 on R^N−1\Ct and its a.c. part has the following density density ψ_k,ℓ,t from [Ser99, Cor. 4.4]

∀(k, ℓ)∈X²,∀y∈R^N−1, ψ_k,ℓ,t(y) := 1_Ct(y)a^N−1 X∞ n=0

e^−at(at)ⁿ

n! (30)

× X

k₁≥0, . . . , k_N−1≥0, P_N−1

j=1 k_j≤n

x^t;y_{n;k1,...,kN−1}p_k,ℓ(n+N, k₁, . . . , k_N−1)

where the non-negative coefficient p_k,ℓ(n+ N, k₁, . . . , k_N−1) satisfies 0 ≤ PN−1

ℓ=1 p_k,ℓ(n+ N, k₁, . . . , k_N−1)≤1 and we have the following relation

X

k₂≥0, . . . , k_N≥0, PN

j=2k_j≤n

n!x^t;y_n;k²₂^,...,y_,...,k^N

N = 1 (31)

For every (k, ℓ) ∈X², the Lebesgue decomposition of the positive measure Fk,ℓ,t defined by P_k{X_t=ℓ, L^′_t∈ ·} writes as follows

∀B ∈B(R^N−1), Fk,ℓ,t(1_B) =P_k{X_t=ℓ, L^′_t∈B}= Z

B

ψ_k,ℓ,t(y)dy+α_k,ℓ,t(1_B). (32) First note that Condition (I-A) is fulfilled since the matrix P := P₁ =e^G is irreducible and aperiodic when the generator G of the driving jump process is irreducible. Second, the random variables kL_tk (and so kY_tk) are bounded, so that the moment condition (Mα) is satisfied for anyα >0. This allows us to apply Theorem 2.1 to derive a CLT for{t^−1/2Y_t^′}t≥0. Next, the main steps of the proof are to check that Conditions(AC1)-(AC2)hold under the assumptions of Proposition 5.1 on the generator G.

5.2.1 Checking Condition (AC2)

For everyy∈R^N−1, we introduce the following realN ×N-matrix Ψt(y) := (ψ_k,ℓ,t(y))_(k,ℓ)∈X2.

where ψ_k,ℓ,t is defined in (30). The useful properties of ψ_k,ℓ,t are given in the next lemma which is proved in Appendix B.

Lemma 5.3 For anyt >0, Ψ_t vanishes on R^d\Ct, is continuous on Ct and differentiable on Ct. We have

sup

t>0

sup

y∈RN−1kΨ_t(y)k0≤sup

t>0

sup

y∈RN−1kΨ_t(y)1^⊤k0≤a^N−1; (33a) j= 1, . . . , N −1 sup

t>0

sup

y∈Ct

∂Ψ_t

∂y_j(y)

0≤2a^N. (33b)

There exists ρ∈(0,1) such that

∀y∈∂Ct, kΨt(y)k0 =O(e^at(ρ−1)(1 +at)). (33c) Now, let us check that Conditions (AC2)hold for the function

G_t(y) = Ψ_t(y+m^′t) (34)

associated with the absolutely continuous part of the probability distribution of the centered r.v. (X_t, Y_t^′) = (X_t, L^′_t−m^′t) under P_k and m^′ := (π₁, . . . , π_N−1). It vanishes on the open convexDt:=T_−m^′_t(Ct) which is the translate of Ct by the translationT_−m^′_t of vector−m^′t.

More precisely, Dt=

y∈R^N−1:j = 1, . . . , N −1, y_j∈(−m_jt,(1−m_j)t),hy,1i< m_Nt . with adherence

Dt=

y ∈R^N−1 :j= 1, . . . , N −1, y_j ∈[−m_jt,(1−m_j)t],hy,1i ≤m_Nt . (35a) and boundary

∂Dt=

N−1[

j=1

y ∈ Dt|y_j =−m_jt

∪

y∈ Dt:hy,1i=m_Nt . (35b)

Then it follows from the basic properties of functionΨ_tonCtstated in Lemma 5.3 that G_tis continuous on Dt and differentiable on Dt with differential

∀j= 1, . . . , N −1, ∀y∈ Dt, ∂G_t

∂y_j(y) = ∂Ψ_t

∂y_j(y+m^′t).

Moreover, we obtain from (33a)-(33c) sup

t>0

sup

y∈RN−1kG_t(y)k0 ≤sup

t>0

sup

y∈RN−1kG_t(y)1^⊤k0 ≤sup

t>0

sup

y∈RN−1kΨ_t(y)1^⊤k0 ≤a^N−1;

∀y∈∂Dt, kG_t(y)k0=kΨ_t(y+m^′t)k0=O 1 t

; sup

t>0

sup

y∈Dt

∂G_t

∂y (y)

0 ≤sup

t>0

sup

y∈Ct

∂Ψ_t

∂y (y)

0 ≤2a^N. 5.2.2 Checking Condition (AC1)

For any i ∈ {1, . . . , N}, we introduce the following (N −1) ×(N −1)-subgenerator of G, G_i^c_i^c := (G(k, ℓ))_k,ℓ∈{i}^c. If G_i^c_i^c is irreducible then ke^{tGic ic}k0 = O(e^−rit) where −r_i is the Perron-Frobenius negative eigenvalue of Gi^ci^c. Thus, if Gi^ci^c is irreducible for any i ∈ {1, . . . , N}

i∈{1,...,N}max ke^{tGic ic}k0 =O(e^−rt) where r:= min

i (r_i)>0. (36)

In a first step, we study the singular partAtof the Lebesgue decomposition (32) whereAt

is theN ×N-matrix with entries in the set of bounded positive measures on R^N−1 At:= (α_k,ℓ,t)_(k,ℓ)∈X2.

We show thatkAt(1RN−1)k0 goes to 0 at a geometric rate when tgrows to infinity. Note that At(1_B) = 0 for every B ∈ B(R^N−1) such that B∩∂Ct =∅. Next, it remains to show that there existc >0and ρ∈(0,1) such that

∀t >0, kAt(1_∂Ct)k0 ≤cρ^t. First, let us consider, for anyi∈ {1, . . . , N−1}, the setd_i,t :=

(y₁, . . . , y_N−1)∈ Ct|y_i= 0 : α_k,ℓ,t(1_di,t) ≤ P_k{L_t(i) = 0}=

(0 if k=i P

ℓe^{tGic ic}(k, ℓ) if k6=i.

Thus, max_ikAt(1_di,t)k0 = O(e^−rt) from (36). Second let us denote s_t :=

(y₁, . . . , y_N−1) ∈ Ct|P_N−1

j=1 y_j =t . We can write for all(k, ℓ)∈X² α_k,ℓ,t(1_st) ≤ P_k{L_t(N) = 0, X_t=ℓ}

≤ P_k{Lt(N) = 0}=

(0 ifk=N P

ℓe^{tGN c N c}(k, ℓ) ifk6=N.

Therefore, kAt(1_st)k0 = O(e^−rt). Combining the previous estimates, we obtain that there existc >0 and ρ∈(0,1) such that

∀t >0, kAt(1RN−1)k0 =kAt(1_∂Ct)k0 ≤cρ^t.

It follows that there existc >0 and ρ∈(0,1) such that

∀t >0, kMt(1RN−1)k0 =kAt(1RN−1)k0≤cρ^t.

where Mt is the matrix associated with the singular part of the probability distribution of (X_t, Y_t^′).

In a second step, we prove that for everyt0 >0 sup

t∈[t0,2t0)kΨb_t(ζ)k0 −→0 when kζk →+∞.

Indeed, for every t > 0, for every (k, ℓ) ∈ X², the Fourier transform ψb_k,ℓ,t of ψ_k,ℓ,t has the following form (ifN = 2 consider only the first integral)

ψb_k,ℓ,t(ζ) = Z _t

0

Z _t−y1

0 · · ·

Z _t−^P_j≤N−2yj

0

ψ_k,ℓ,t(y)e^{iζN−1yN−1}dy_N−1. . . dy₂dy₁. Using an integration by part, we obtain that for every ζ_N−1 6= 0

Z t−P

j≤N−2yj

0

ψ_k,ℓ,t(y)e^{iζN−1yN−1}dyN−1= 1

iζ_N−1 h

ψ_k,ℓ,t(y)e^{iζN−1yN−1}iyN−1=t−P

j≤N−2yj

yN−1=0 −

Z _t−^P_j≤N−2yj

0

∂ψ_k,ℓ,t

∂y_N−1(y)e^{iyN−1ζN−1}dy_N−1

!

so that, we obtain from the bounds (33a)-(33b) that

|ψb_k,ℓ,t(ζ)| ≤ 1

|ζ_N−1| Z t

0 · · · Z t−P

j≤N−3yj

0

2a^N−1+ 2a^N(t− X

j≤N−2

y_j)

dy_N−2. . . dy₁

≤ 1

|ζN−1| 2a^N−1 t^N−2

(N −2)!+ 2a^N t^N−1 (N −1)!

.

Finally, for everyt0 >0 sup

t∈[t0,2t0)kΨbt(ζ)k0 ≤ 2a^N−1(1 +a) max(1, t^N₀ )

|ζ_N−1| .

For each j ∈ {1, . . . , y_N−2}, using similar computations from an integration by parts with respect to the variabley_j, the same inequality holds with ζ_j in place of ζ_N−1. Therefore, we obtain that

∀ζ ∈R^N−1\{0}, sup

t∈[t0,2t0)kΨb_t(ζ)k0 ≤ 2a^N−1(1 +a) max(1, t^N₀ ) kζk0

.

and this quantity goes to 0 when kζk →+∞. Since G_t(y) = Ψ_t(y+tm) we have Gb_t(ζ) = e^{−ihm′,ζit}Ψb_t(ζ) for anyζ ∈R^N−1 and

∀t >0, ∀ζ ∈R^N−1, kGb_t(ζ)k0=kΨb_t(ζ)k0. It follows that

Γ(ζ)≡Γ_t0(ζ) := sup

t∈[t0,2t0)kGb_t(ζ)k0 −→0 whenkζk0→+∞.

6 A uniform LLT with respect to transition matrix P

Let P denote the set of irreducible and aperiodic stochastic N ×N-matrices. The topology in P is associated (for instance) with the distance d(P, P^′) := kP −P^′k∞ (P, P^′ ∈ P).

Again (X_t, Y_t)_t∈T is a centered MAP with state space X×R^d, where X := {1, . . . , N}, and {P_t}t∈T denotes the transition semi-group of the Markov process{X_t}t∈T. In this section, the stochastic matrix P := P₁ is assumed to belong to a compact subset P0 of P, and we give assumptions for the LLT of Theorem 2.2 to hold uniformly in P ∈ P0.

For every k∈Xthe underlying probability measure P_kand the associated expectation E_k depend onP. To keep in mind this dependence, they are denoted byP^P

k andE^P

k respectively.

Similarly we use the notations Σ^P, Y^Pt,G^Pt ,M^Pt and G^P_t for the covariance matrix of Theo- rem 2.1 and the matrices in (6a)-(6b) respectively. LetMdenote the space of the probability measures onR^d equipped with the total variation distance d_{T V}.

LetP0 be any compact subset ofP. Let us consider the following assumptions:

U 1 : For any(k, ℓ)∈X², the map P 7→ Y^Pk,ℓ,1 is continuous from (P0, d) into (M, d_{T V}).

U 2 : There exist positive constantsα andβ such that

∀P ∈ P0, ∀ζ ∈R^d, αkζk² ≤ hζ,Σ^Pζi ≤βkζk². (37) U 3 : The conditions (M3),(AC1) and (AC2) hold uniformly in P ∈ P0.

Theorem 6.1 Under Assumptions (U1)-(U3), for every k ∈ X, the density f_k,t^P (·) of the a.c. part of the probability distribution of t^−1/2Y_t under P^P

k satisfies the following asymptotic property when t→+∞:

sup

P∈P0

sup

y∈R^d

f_k,t^P (y)−η

ΣP(y)=O(t^−1/2) +O sup

P∈P0

sup

y /∈Dt

ηΣP(t^−1/2y) .

Assumptions (U3) read as follows:

U3 -1 : The r.v.{Yv}v∈(0,1]∩T satisfies the following moment condition:

M := sup

P∈P0

maxk∈X sup

v∈(0,1]∩T

E^P

k[kY_vk³]<∞. (38)

U3 -2 : There exist c >0 and ρ∈(0,1) such that

∀t >0, sup

P∈P0

kM^Pt (1R^d)k0≤cρ^t (39) and there exists t₀ >0 such that

ρ^t0max(2, cN)≤1/4 (40a)

Γ_t0(ζ) := sup

P∈P0

sup

w∈[t0,2t0)kGc^P_w(ζ)k0 −→0 whenkζk →+∞. (40b)

U3 -3 : For any t > 0, there exists an open convex subset Dt of R^d such that G_t vanishes on R^d\ Dt, where Dt denotes the adherence of Dt. Moreover G_t is continuous on Dt and differentiable on Dt, with in addition

sup

P∈P0

sup

t>0

sup

y∈Dt

kG^P_t (y)k0 <∞ (41a)

sup

P∈P0

sup

y∈∂Dt

kG^P_t (y)k0 =O 1 t

where ∂Dt:=Dt\ Dt (41b)

j= 1, . . . , d: sup

P∈P0

sup

t>0

sup

y∈Dt

∂G^P_t

∂y_j (y)

0 <∞. (41c)

Proof. The proof of Theorem 6.1 borrows the same way as for Theorem 2.2. What we only have to do is to prove that the bounds in Lemmas 3.1-3.3 are uniform in P ∈ P0 under Assumptions (U1)-(U2). For Lemma 3.3, this is obvious by using (U3-2). The proof of Theorem 6.1 will be complete if we establish the uniform version of Lemma 3.1 and 3.2. This

is done below.

Lemma 6.1 Assume that Condition (U1) holds and that, for some α >0, M_α:= sup

P∈P0

maxk∈X sup

v∈(0,1]∩T

E^P

k[kY_vk^α]<∞. Then the map (P, ζ)7→Yc^P1(ζ) is continuous from P0×R^d intoMN(C).

Proof. We may suppose that α ∈(0,1]. Let (k, ℓ) ∈X²,(P, P^′)∈ P²0 and (ζ, ζ^′)∈R^d×R^d. To simplify we write Y andY^′ for Y^Pk,ℓ,1 and Y^Pk,ℓ,1^′ . Then

cY^P1(ζ)

k,ℓ− Yd^P1^′(ζ^′)

k,ℓ

= Z

R^d

e^ihζ,yiY(dy)− Z

R^d

e^ihζ′,yiY^′(dy)

≤ Z

Rd

e^ihζ,yi−e^ihζ′,yiY(dy)

+ Z

Rd

e^ihζ′,yiY(dy)− Z

Rd

e^ihζ′,yiY^′(dy)

≤ 2kζ −ζ^′k^α Z

Rdkyk^αY(dy) +d_{T V}(Y,Y^′)

≤ 2M_αkζ −ζ^′k^α+d_{T V}(Y,Y^′)

(use |e^iu −1| ≤ 2|u|^α, u ∈ R, and the Cauchy-Schwarz inequality to obtain the second inequality above). The desired continuity property then follows from (U1).

From now on, we sometimes omit the notational exponentP. The uniformity in Lemma 3.1 is obtained as follows. Recall that, for any P ∈ P0 fixed, Formula (15) with t = n ∈ N follows from the standard perturbation theory. Similarly, using Lemma 6.1, Formula (15) witht=n∈N can be obtained for everyP ∈ U0 and for allζ ∈R^d such thatkζk ≤δ, with δ > 0 independent from P₀ ∈ P0 since P0 is compact. Moreover the associated functions λ(·), L_k,t(·) and R_k,t(·) in (15) (depending on P) satisfy the properties (16a)-(16d) in a uniform way in P ∈ P0 from (U3-1). Note that Condition (U2) is useful to obtain (16b) uniformly in P ∈ P0. The passage from the discrete-time case to the continuous-time again follows from [FHL12, Prop. 4.4] since the derivatives of ζ 7→Ycv(ζ) are uniformly bounded in (P, v)∈ P0×(0,1]from (U3-1).