Ergodicity of a certain class of non Feller models : applications to <span class="mathjax-formula">$\textit {ARCH}$</span> and Markov switching models

May 2004, Vol. 8, p. 76–86 DOI: 10.1051/ps:2004003

ERGODICITY OF A CERTAIN CLASS OF NON FELLER MODELS:

APPLICATIONS TO ARCH AND MARKOV SWITCHING MODELS

^∗,∗∗

Jean-Gabriel Attali

¹

Abstract. We provide an extension of topological methods applied to a certain class of Non Feller Models which we call Quasi-Feller. We give conditions to ensure the existence of a stationary distribu- tion. Finally, we strengthen the conditions to obtain a positive Harris recurrence, which in turn implies the existence of a strong law of large numbers.

Mathematics Subject Classification. 60B05, 60B10, 60J10.

Received November 7, 2002. Revised June 12, 2003.

Introduction

It is well-known that to ensure the existence of invariant probability measure for a Feller Markov chain with a Polish state space S, that is whose transition operator maps the span of bounded and continuous function from S to R into itself, it suffices to establish the tightness of the sequence of probability measures (¹_nn−1

k=0P^k(x₀,dy))_n∈N^∗ for a certain x₀ in S (see e.g. [2] or [5]). This elementary approach is very well adapted to the problem of Markov chains inR^d. For instance, the existence of at least one invariant probability measure for a nonlinear regression modelNAR of the formX_t+1=F(X_t) +ε_t+1, whereX_ttakes values inR^d and the ε_t are i.i.d., can be easily proven providing that F(x) is a continuous function, to ensure that the transition probability is Feller, and that it grows slowly whenxgoes to infinity, to ensure the tightness of the sequence (_n¹_n−1

k=0P^k(x,dy))_n∈N∗. For the more general modelARCHof the formX_t+1=F(X_t) +G(X_t)ε_t+1, it suffices that F(x) and G(x) be continuous functions and have a good behavior when x goes to infinity.

Unfortunately, this topological approach doesn’t work for non Feller models. For instance, considering aARCH model, wheneverF orGpossess a point of discontinuity it is not possible to apply this method.

However, via techniques of petite sets (see e.g. [3] or [5]), we know that someNAR models, where F is a threshold function, have invariant probability although they are not Feller. Thus, our aim is to extend the topological approach to the study of non Feller models as well. In this context, we introduce the concept of Quasi-Feller models.

The paper is organized as follows. In Section 1 we define Quasi-Feller models. In Section 2 we then give conditions to obtain at least one invariant probability measure and finally, in Section 3 we closely study the

Keywords and phrases.Ergodic, Markov chain, Feller, Quasi-Feller, invariant measure, geometric ergodicity, rate of convergence, ARCH models, Markov switching.

∗To my Father.

∗∗ Special thanks to Gilles Pag`es for helpful comments on this work.

1 ENSAE, Timbre J120, Bureau E01, 3 avenue Pierre Larousse, 92245 Malakoff Cedex, France; e-mail:[email protected] c EDP Sciences, SMAI 2004

problem of the positive Harris recurrence for such models. ARCH models and nonlinear autoregressive models with Markov switching (NAR-MS) illustrate our results throughout the paper.

1. Definition of Quasi-Feller models 1.1. Definitions

(S, d) is a separable complete metric space andB(S) is its Borel σ-field. We denote the class of bounded continuous functions fromS toRbyC_b(S,R). Let us first define the Quasi-Feller models as follows:

Definition 1.1. A transition kernel P(x,dy) is Quasi-Feller if there exist a topological space W, a Borel functionH :S→W such that, for all compact subsetK,H(K) is a compact subset and a family of probability measures (Q(w,dy))_w∈W on (S,B(S)), verifying:

i) the kernelQdefined for any measurable and bounded functiong:S→RbyQg(w) :=

Sg(y)Q(w,dy) is Feller that is:

g∈C_b(S,R)⇒Qg∈C_b(W,R);

ii) ∀g measurable mapping fromS toR, P g(x) =Qg(H(x));

iii) ∀w∈

K∈KH(K) we haveQ(w, D_H) = 0, whereD_H denotes the set of discontinuity points ofH and K is the span of compact subsets ofW.

Remark 1.2.

• Clearly

K∈KH(K)⊂H(S). IfH is a continuous function, the above inclusion holds as an equality.

On the other hand, as soon as H is not continuous at one point, the inclusion may be strict.

• D_H is a measurable set (seee.g. [1]).

• The decomposition ofP followingQandH is not unique.

• A Feller transition kernelP is Quasi-Feller with W =S,H =Id_S,D_H =∅ andQ=P.

• Qis a transition probability if and only ifW =S.

• In our examples,W generally will beR^p or one of its subsets.

1.2. Examples

1.2.1. ARCH model

Let F : R^d → R^d and G : R^d → S⁺(d,R) (where S⁺(d,R) denotes the set of definite positive matrix) be measurable functions. Let (ε_t)_t≥1 be a white noise whose distribution µ is defined on a probability space (Ω,A,P). TheARCH model is then defined by:

X_t+1=F(X_t) +G(X_t)ε_t+1, X₀∈R^d. (1) Let us define the function:

G⁽⁻¹⁾ : R^d→S⁺(d,R), x→(G(x))⁻¹, and the norm of ad×dmatrix:

A:= sup

|u|=1

|Au|

|u| · The transition probability P(x,dy) of theARCH is given by:

∀g bounded measurable function, P g(x) =

g(F(x) +G(x)y)µ(dy). (2)

Proposition 1.3. If the set D_F,G:=D_F∪D_G satisfies:

∀ w¹, w²

∈

K∈K

F(K)×G(K), µ w²₋₁

D_F,G−w¹ = 0, (3)

then the ARCH model is Quasi-Feller.

Proof. Let us define, for any bounded measurable function g:R^d→Rthe function Qgas follows:

∀ w¹, w²

∈R^d×S⁺(d,R), Qg w¹, w²

=

g

w¹+w²y

µ(dy). (4)

We have P g(x) = Qg(F(x), G(x)). Q obviously is Feller according to the Definition 1.1 (Q is no longer a transition probability becauseE=R^d×S⁺(d,R)).

The hypothesis∀w∈

K∈KF(K)×G(K), Q(w, D_F,G) = 0 turns into

∀w:=

w¹, w²

∈

K∈K

F(K)×G(K), µ w²₋₁

D_F,G−w¹ = 0.

Remark 1.4. In this case, we have: W =R^d×S⁺(d,R), andG(K)⊂S⁺(d,R).

1.2.2. Nonlinear autoregressive model with Markov switching (NAR-MS) Definition 1.5. (Y_t)_t∈N∈

R^d_N

is aNAR-MS if:

Y_t:=f_Xt(Y_t−1) +ε_t,

where (X_t)_t≥0 is a positive Harris recurrent chain on {1,· · ·, m}, for all k ∈ {1,· · ·, m}, f_k is a measurable function fromR^d into itself and theε_tare i.i.d. random variables taking values inR^d.

Example 1.6. Let (Y_t)_t∈N be a NAR-MS. Let µ denote the distribution of ε₀ and D_k denote the set of discontinuity points off_k. If:

∀k∈ {1,· · ·, m}, ∀z∈

K∈K

f_k(K), µ(D_k−z) = 0,

then theNAR-MS is Quasi-Feller.

Proof. X_t being positive Harris, there exist a measurable function h : {1,· · ·, m} ×R → {1,· · · , m} and a sequence of i.i.d. real valued variables (η_t)_t≥1 with distribution ν and which are independent from the process (ε_t)_t≥0 such that:

X_t=h(X_t−1, η_t). Then, the process:

Z_t= X_t

Y_t

=

h(X_t−1, η_t) f_{h(Xt−1,ηt)}(Y_t−1) +ε_t

, is a Markov chain whose transition kernelP verifies:

∀g∈C_b

{1,· · ·, m} ×R^d,R

, P g(x, y) =

Rν(dη)

R^dg

h(x, η), f_h(x,η)(y) +ε µ(dε).

Now let define, for allg∈C_b

{1,· · ·, m} ×R^d,R

the function Qgas follows:

∀(x, z₁,· · ·, z_m)∈ {1,· · ·, m} × R^d_m

,

Qg(x, z₁,· · ·, z_m) =

Rν(dη)

R^dg

h(x, η), m k=1

1

{k}(h(x, η))z_k+ε

µ(dε). (5)

For all x and η, (z₁,· · · , z_m) → g(h(x, η),_m

k=1

1

{k}(h(x, η))z_k +ε) is a continuous function. So, the func- tion (z₁,· · ·, z_m) → Qg(x, z₁,· · · , z_m) is continuous. But x takes values in a finite set, hence the function (x, z₁,· · · , z_m)→Qg(x, z₁,· · ·, z_m) is also continuous.

Setting H(x, y) = (x, f₁(y),· · ·, f_m(y)), we have:

∀g∈C_b

{1,· · ·, m} ×R^d,R

, P g(x, y) =Qg(H(x, y)). To finish,D_H⊂ {1,· · ·, m} ×_m

k=1D_k. Hence, the hypothesis:

∀k∈ {1,· · · , m}, ∀z_k∈

K∈K

f_k(K), µ(D_k−z_k) = 0,

implies that:

∀(x, z₁,· · ·, z_m)∈

K∈K

H(K), Q(x, z₁,· · ·, z_m, D_H) = 0,

and so theNAR-MS is Quasi-Feller.

2. Existence of a stationary probability measure for Quasi-Feller models

Theorem 2.1. Let us consider a Quasi-Feller transition probabilityP(x,dy)associated to a functionH a kernel Qaccording to Definition 1.1.

Suppose there existsx₀∈S such that (_n¹_n−1

k=0P^k(x₀,dy))_n∈N^∗ is tight. Then (a) the model has an invariant probability measure;

(b) for all invariant probability measureν and all measurable set C satisfying:

∀w∈

K∈K

H(K), Q(w, C) = 0, (6)

thenν(C) = 0. In particular,ν(D_H) = 0.

Proof. The idea is to show, as for Feller Models, that any limit point ν for weak convergence of the tight sequence _n¹_n−1

k=0P^k(x₀,dy) is an invariant probability distribution.

The key is to show thatν(D_H) = 0. Let us prove directly point (b). Let C be a measurable subset of S satisfying (6). Let prove thatν(C) = 0.

IfF be a closed subset ofC. thenF also satisfies (6), i.e.:

∀w∈

K∈K

H(K), Q(w, F) = 0.

Let (h_p)_p∈N be the sequence of functions fromS to Rdefined by:

∀p∈N^∗, ∀x∈S, h_p:= 1−min (pd(x, F),1).

For allp∈N,h_pare continuous functions (evenp-Lipschitz). Moreover, they verifyh_p∞≤1 andh_p

1

_F ⁼

1

F forF is a closed subset.

Let us consider now (Qh_p)_p∈Nthe sequence of functions fromW toR. ForQis Feller,Qh_pare so continuous functions. On the other hand, Q(w,dy) is a probability measure for all w ∈ W, hence following dominated convergence Theorem, we have:

∀w∈

K∈K

H(K), Qh_p(w)Q(w, F).

Let K be a compact subset of S. Qh_p 0 on H(K) for Q(w, F) = 0 if w ∈ H(K). At last, H(K) is, by hypothesis, a compact subset. Then, the Dini’s Theorem implies that the above convergence is uniform onH(K) i.e.

sup

w∈H(K)

Qh_p(w)→0 whenp→+∞. (7)

Define ν_n := _n¹_n−1

k=0P^k(x₀,dy) with the usual conventionP⁰(x₀,dy) = δ_x0(dy). The sequence (ν_n)_n∈N^∗ is tight, so we have:

∀ε >0, ∃Kεcompact subset ofSsuch that, ∀n∈N^∗, ν_n(^cK_ε)≤ε. (8) It follows that, for all n,p≥1,

ν_n(P h_p) =

Kε

P h_pdν_n+

cKε

P h_pdν_n≤ sup

y∈Kε

P h_p(y) +ν_n(^cK_ε)≤ sup

y∈Kε

P h_p(y) +ε.

ButP h_p=Qh_p◦H hence, for alln,p≥1,

ν_n(P h_p)≤ sup

w∈H(Kε)

Qh_p(w) +ε.

Consider a subsequence ν_ϕ(n)

n∈N^∗ that converges weakly to a probability measureν. Because for all bounded functionf we have:

|ν_n(P f)−ν_n(f)| ≤ 2f_∞ n , it follows immediately that:

ν(h_p) = lim

n→∞ν_ϕ(n)(h_p) = lim

n→∞ν_ϕ(n)(P h_p)≤ sup

w∈H(Kε)

Qh_p(w) +ε.

Thus, by (7) we have:

∀ε >0, lim sup

p→∞ ν(h_p)≤ε, showing that, ν(F) = lim_p→∞ν(h_p) = 0.

Then, for all closed subsetF ofC we haveν(F) = 0.By regularity of the probability measureν, we have:

ν(C) := sup{ν(F), F ⊂C, F closed subset} (see [1]).

Finallyν(C) = 0. In particular, we haveν(D_H) = 0.

Consider nowh∈C_b. One clearly hasP h=Qh◦H is a continuous function on the continuity set ofH. So, P his ν-a.s.continuous. Then:

ν_ϕ(n)(P h)^n→+∞−→ ν(P h) forν_ϕ(n)⇒ν.

But

|ν_n(P h)−ν_n(h)|^n→+∞−→ 0 andν_ϕ(n)(h)^n→+∞−→ ν(h),

so we can deduce thatν(P h) =ν(h) and to finishνP =ν because his an arbitrary function in C_b. To ensure the tightness of (_n¹_n−1

k=0P^k(x,dy))_n∈N∗ we usually use a Lyapunov function V i.e. verifying fora∈S:

d(x,a)→∞lim V(x) = +∞.

Let recall the following result:

Proposition 2.2. Let P(x,dy)be a transition probability on S andV be a continuous Lyapunov function.

Pakes’ lemma [6]suppose that:

i)∀x∈S, P V(x)≤V(x) +β;

ii) for a∈S,lim_d(x,a)→∞(P V −V)(x) =−∞.

Then, for all xinS the sequence(¹_nn−1

k=0P^k(x,dy))_n∈N^∗ is tight.

3. Irreducibility and positive Harris recurrence

The aim of this section is to extend the criterion of positive Harris recurrence from the Feller case as presented in [5] to our Quasi-Feller models. Anyψ-irreducible Feller chain on a polish space S, where the support ofψ has nonempty interior, is such that every compact subset ofS is petite. If we add the hypothesis of tightness, for all xin S, of the sequence of probability measures (_n¹_n−1

k=0P^k(x,dy))_n∈N it is easy to show the chain is positive Harris recurrent (seee.g.[5]). Our aim is to extend this result to Quasi-Feller chain.

3.1. Definitions

Let recall first the notion ofφ-irreducibility as defined in [4]:

Definition 3.1. Consider a Markov chain with transition kernelP(x,dy) onS andφa nonnegative measure.

The chain is said to beφ-irreducible if,

∀x∈S, φ(dy)

+∞

n=1

a_nPⁿ(x,dy),

where thea_n are nonnegative real numbers such that_+∞

n=1a_n= 1.

Definition 3.2. We will say an homogeneous Markov chain (X_t)_t∈Nis positive Harris recurrent if there exists a probability measureν such that:

∀g:S→Rbounded continuous function, ∀x∈S, 1 n

n−1

k=0

g(X_k)^Px−→-a.s.ν(g).

This definition clearly implies that ν is the unique invariant probability measure of the chain. Moreover, if a chain is positive Harris recurrent chain withν for unique stationary distribution, then it is ν-irreducible (see [5]).

In the following, ψ will denote a maximal measure of irreducibility, i.e. a measure of irreducibility that dominates all measures of irreducibility. The existence of such a measure for a φ-irreducible Markov chain is proved in [5].

Definition 3.3. Let P(x,dy) be the transition kernel of a ψ-irreducible Markov chain. A nonempty set A∈ B(S) is called petite if it verifies:

x∈Ainf K_a(x,dy)≥δψ(dy), whereK_a(x,dy) :=_+∞

k=1a_kP^k(x,dy) and_+∞

k=1a_k = 1, ∀k≥1, a_k>0, δ >0.

Remark 3.4.

• K_aK_b(x,dy) :=

K_a(x,dz)K_a(z,dy) =K_a∗b(x,dy).

• The property of petiteness depends only onP(x,dy) but not on the chosen sequence (a_n)_n∈N∗.

• Every subset of a petite set is petite.

3.2. Positive Harris recurrence of Quasi-Feller models

To obtain positive Harris recurrence for our models, the key is to prove the following result:

Proposition 3.5. LetP(x,dy) =Q(H(x),dy)be aψ-irreducible Quasi-Feller transition probability. Ifsupp{ψ}

has nonempty interior, then all compact subsets of S are petite.

Proof. The proof relies on the following two Lemmas:

Lemma 3.6. Let P(x,dy) =Q(H(x),dy)be a Quasi-Feller. Then:

(a) for any bounded function f : S → R which is continuous on S\D_H, and for any k ∈ N^∗, P^kf is continuous on S\DH;

(b) ∀F closed set, ∀k∈N^∗, x→P^k(x, F)is upper semicontinuous on S\DH; (c) ∀O open set,∀k∈N^∗, x→P^k(x, O)is lower semicontinuous onS\DH.

Consequently K_a(x,dy)is a Quasi-Feller transition probability. More precisely, there exists a Feller transition kernel K˜_a(x,dy)such thatK_a(x,dy) = ˜K_a(H(x),dy).

Proof. Let us show that iff is a lower semicontinuous function onS\DH, so isP f.

Let f be a lower semicontinuous function on S\DH. Then f is the limit on S\DH of a growing se- quence (g_p)_p∈N of bounded Lipschitz functions onS\DH.

During the proof of Theorem 2.1, we showed if g : S → R is a bounded continuous function then P g is continuous onS\D_H.

So we have:

P g(x_n)→P g(x) when x_n→x.

Thus, we can write:

∀x∈S\D_H, P(x_n,dy)⇒P(x,dy) whenx_n→x.

Then for anyP(x,dy)-a.s.continuous functiong:

P g(x_n)→P g(x) six_n→x. (9)

ButP(x, D_H) = 0 by hypothesis. ConsequentlyP f is P(x,dy)-a.s. continuous wheneverf is a bounded and continuous function onS\DH.

It shows that for allp∈N,P g_pis continuous onS\D_H. Then,x→P f(x) is a lower semicontinuous function onS\D_H as a growing limit of continuous functions onS\D_H. So, Point (a) is true. Points (b) and (c) follow easily.

To see the second part of the Lemma, let g:S→Rbe a bounded continuous function. We have:

K_ag(x) =

+∞

k=1

a_kP^kg(x)

=

P(x,dy)

+∞

k=1

a_kP^k−1g(y)

=

Q(H(x),dy)

+∞

k=1

a_kP^k−1g(y).

By point (a), we know that _+∞

k=1a_kP^k−1g(y) is continuous on S\DH and so it is Q(z,dy)-a.s. continuous.

Q being Feller, the function f(z) =

Q(z,dy)_+∞

k=1a_kP^k−1g(y) is a bounded continuous function from W to R. It implies that ˜K_a(z, .) =

Q(z,dy)_+∞

k=1a_kP^k−1(y, .) is Feller which proves the announced result for

K˜_a(H(x), .) =K_a(x, .).

The following lemma and the end of the proof of Proposition 3.5 are extensions of Lemma 6.2.7 and Propo- sition 6.2.8 from [5].

Lemma 3.7. LetP(x,dy) =Q(H(x),dy)be a Quasi-Feller transition probability and letAbe a Borel set such that A∩(S\DH) is a petite set forP. ThenA∩(S\DH)is a petite set for P.

Proof. By hypothesis, we have inf_{x∈A∩(S\DH}K_a(x,dy)≥δψ(dy),withδ >0.

LetB∈ B(S) and letF ⊂Bbe a closed set. Lemma 3.6 permits us to ensure that the functionx→K_a(x, F) is a uniform limit of functions that are upper semicontinuous onS\DH. So it is a function that is also upper semicontinuous onS\DH. Thus:

x∈A∩(S\Dinf H)K_a(x, F) = inf

x∈A∩(S\DH)K_a(x, F).

F being a subset ofB, we can deduce:

x∈A∩(S\Dinf H)K_a(x, B)≥δψ(F).

Forψis a regular measure,

x∈A∩(S\Dinf H)K_a(x, B)≥δψ(B),

which proves that A∩(S\DH) is a petite set.

End of the proof of Proposition 3.5: The chain beingψ-irreducible, there exists a petite setA of strictly positiveψ-measure (see [5]).

Let us define for allk∈N^∗, the following sets:

A_k:=

x∈S/ K_a(x, A)> 1 k

·

It is clear that:

k∈N^∗

A_k ={x∈S/ K_a(x, A)>0}=S for ψ(A)>0.

So, there exists k₀∈N^∗ such that for allk≥k₀,A_k=∅. Fork≥k₀ andx∈A_k, we have:

K_a∗a(x, dy) =

K_a(x,dz)K_a(z,dy)

≥

A

K_a(x,dz)K_a(z,dy)

≥δψ(dy)×K_a(x, A)

≥ δ kψ(dy).

Thus, fork≥k₀,A_k is a petite set of strictly positiveψ-measure.

LetO:=

◦

supp{ψ}. O is a nonempty set and is of strictly positiveψ-measure. Thus, for allk≥k₀, A_k∩O is of strictly positiveψ-measure. On the other hand,A_k∩O is a petite set because it is a subset of the petite set A_k. SoA_k∩O is a petite set of strictly positiveψ-measure. By Proposition 2.1, we know thatν(D_H) = 0, so ψ(D_H) = 0 forψν.Then, for allk≥k₀,A_k∩O∩(S\D_H) is a petite set of strictly positiveψ-measure.

Lemma 3.7 implies that for allk≥k₀, the sets:

B_k :=A_k∩O∩(S\D_H), are petite set of strictly positiveψ-measure.

Let us define, for all k≥k₀, the setsF_k :=A_k∩O. F_k are closed sets and verify

k≥k0F_k =O. Hence, at least one of theF_k is nonempty. By Baire’s Theorem, we know then that at least one of theF_k has a nonempty interior.

LetF_k1 (k₁≥k₀) be one of theF_k with a nonempty interior and letO_k1 be its interior. O_k1∩(S\D_H) has strictly positive ψ-measure because O_k1 ⊂ supp{ψ} and is a petite set as subset of B_k1. Hence there exists δ₁>0 such that:

x∈Ok1inf∩(S\DH)K_a(x,dy)≥δ₁ψ(dy).

Letx∈S andB∈ B(S). Let F be a closed subset ofB. We have:

K_a∗a(x, F) =

K_a(x,dy)K_a(y, F)

≥

Ok1∩(S\DH)K_a(x,dy)K_a(y, F)

≥δ₁ψ(F)K_a(x, O_k1∩(S\DH)).

ButK_a(x, D_H) = 0 soK_a∗a(x, F)≥δ₁ψ(F)K_a(x, O_k1).By taking the supremum over the closed sets included in B, we obtain:

K_a∗a(x, B)≥δ₁ψ(B)K_a(x, O_k1). (10)

If we choosea <1 anda_k= (1−a)a^k−1 for allk≥1, we have

K_a(x, O_k1) =

+∞

k=1

(1−a)a^k−1P^k(x, O_k1)

= (1−a)P(x, O_k1) +a

+∞

k=1

(1−a)a^k−1

P(x,dy)P^k(y, O_k1)

= (1−a)P(x, O_k1) +a

P(x,dy)K_a(y, O_k1)

= (1−a)P(x, O_k1) +a

Q(H(x),dy)K_a(y, O_k1)

≥a

Q(H(x),dy)K_a(y, O_k1).

We already know thatK_a(y, O_k1) is lower semicontinuous onS\D_H. Hence,

Q(w,dy)K_a(y, O_k1) is a positive lower semicontinuous on W which implies that its infimum on any compact subset of W is also positive. By hypothesis, for allKcompact subset ofS,H(K) is a compact subset ofW. So we have:

x∈Kinf K_a(y, O_k1)≥a inf

w∈H(K)

Q(w,dy)K_a(y, O_k1) =α >0, what replaced in (10) gives:

x∈Kinf K_a∗a(x,dy)≥αδ₁ψ(dy),

which proves that every compact subset KofS is petite.

Theorem 3.8. Let P(x,dy) =Q(H(x),dy)be the transition kernel of a Quasi-Feller chain. Suppose that:

i) ∀x∈S, (_n¹_n−1

k=0P^k(x,dy))_n∈Nis tight;

ii) the chain is ψ-irreducible and supp{ψ} has a nonempty interior.

Then the chain is positive Harris recurrent.

Proof. Proposition 3.5 proves that we are under assumptions of Theorem 6.2.5 of [5]. Then, the result follows

from Theorem 9.2.2 of [5].

3.3. Applications

Example 3.9. Consider aARCH model. Recall thatµdenotes the distribution ofε₀. Let suppose:

i) µ∼λ_d;

ii) F andGare Riemann integrables;

iii) F,GetG^∗ are bounded on compact sets;

iv) ∀x∈R^d, 1

n

_n−1

k=0P^k(x,dy)

n∈Nis tight.

Then the model is positive Harris recurrent.

Proof. For F andGare Riemann integrable, we haveλ_d(D_F,G) = 0, so the model is Quasi-Feller. The result follows then from Theorem 3.8 for i) implies that the transition probability isλ_d-irreducible.

Example 3.10. Consider (Y_t)_t∈N aN AR-M S. Recall thatµdenotes the distribution ofε₀and suppose that:

i) µ∼λ_d;

ii) thef_k are Riemann integrables;

iii) ∀x∈R^d, (_n¹_n−1

k=0P^k(x,dy))_n∈Nis tight.

Then (Z_t)_t≥0:= (X_t, Y_t)_y≥0is positive Harris.

Proof. For the f_k are Riemann integrable, we have, for allk∈ {1,· · ·, m},λ_d(D_k) = 0, what proves that the model is Quasi-Feller. Again, the result follows then from Theorem 3.8 for the transition probability of (Z_t)_t≥0 is a α⊗λ_d-irreducible chain, whereαis the measure on{1,· · ·, m}such that, for allk,α{k}= 1.

References

[1] P. Billingsley,Convergence of probability measures. John Wiley and Sons, New York (1968) 253.

[2] M. Duflo,M´ethodes R´ecursives Al´eatoires. Techniques Stochastiques, Masson, Paris (1990) 359.

[3] M. Duflo, Algorithmes Stochastiques.Math. Appl.23(1996) 319.

[4] T.E. Harris, The existence of stationnary measures for certain markov processes. Proc. of the 3rd Berkeley Symposium on Mathematical Statistics and Probability2(1956) 113–124.

[5] S.P. Meyn and R.L Tweedie,Markov Chains and Stochastic Stability. Springer-Verlag (1993) 550.

[6] A.G. Pakes, Some conditions for ergodicity and recurrence of markov chains.Oper. Res.17(1969) 1048–1061.