Central Limit Theorem for Random Dynamical System with Jumps and State-Dependent Jump Intensity

Piecewise-deterministic Markov processes (PDMPs) represent a class of stochastic models, that have recently received extensive research attention. They were introduced by Davis [7] as a general and various practical systems in which randomness is limited to jumps. These processes are governed by deterministic semiflows and mentioned jumps which change the state as well the semiflows which will determine evolution of the system. PDMPs have proven to be effective mathematical models for many phenomena, such as gene expression [14] and population dynamics [1]. Results concerning ergodicity and asymptotic stability of these systems are already quite extensively researched [1, 2, 3, 8, 9, 5, 16].

Having the asymptotic stability of such processes, natural questions arise regarding the limit theorems. In [11], we focused on demonstrating a law of the iterated logarithm for some PDMPs in which the intensity of jumps depends on the state of the system. In this paper we prove the central limit theorem (CLT) for this class of processes.

In [4], a criterion for the CLT was introduced, and an example of a model with a constant intensity of jumps was given. The difference between our model and the one considered in [4] lies in the assumption that the intensity of jumps follows an exponential distribution with a constant intensity parameter. This assumption seems to be too restrictive concerning biological models, such as gene expression, in which the relation between the intensity of jumps and the state of the system is visible. It turns out that the criterion for the CLT from [4] is also applicable in the case we are considering.

The paper contains three sections. In Section 2, basic definitions and notations are introduced. In Section 3, we start with an intuitive description of the considered model, and then we formulate strict definitions. We also provide conditions that were used in [6] to show exponential ergodicity. The last section contains the proof of the CLT for the model described earlier, and it is strongly influenced by the methods used in [4].

Let ℝ be the set of all real numbers and let ℝ₊ = [0, ∞). Let (E, ρ) be a Polish space with the σ-field ℬ(E) of all Borel subsets of E. By B(E) we denote the space of all bounded, Borel measurable functions f : E → ℝ, equipped with the supremum norm and we list two of its subsets: C(E) and Lip(E) consisting of all continuous functions and Lipschitz-continuous functions, respectively. A continuous function V : E → ℝ₊ is called Lyapunov function if it is bounded on bounded sets and for some x₀ ∈ E limρ(x,x0)→∞V(x)=∞. \mathop {\lim}\limits_{\rho \left({x,{x_0}} \right) \to \infty} V\left(x \right) = \infty.

Let ℳ_s(E) be the set of all finite, countably additive functions on ℬ(E). By ℳ₊(E) and ℳ₁(E) we denote the subsets of ℳ_s(E) consisting of all non-negative measures and all probability measures, respectively. We write ℳ1V(E) {\cal M}_1^V\left(E \right) for the set of all μ ∈ ℳ₁(E) satisfying ∫_E V (x) μ(dx) < ∞.

The set ℳ_s(E) will be considered with the Fortet-Mourier norm ([12, 13]), given by ‖μ‖FM=sup{|〈f,μ〉|:f∈ℱFM(E)}for μ∈ℳs(E), \matrix{{{{\left\| \mu \right\|}_{FM}} = \sup \{|\left\langle {f,\mu} \right\rangle |:f \in {{\cal F}_{FM}}\left(E \right)\}} \hfill & {{\rm{for}}\,\,\mu \in {{\cal M}_s}\left(E \right)} \hfill \cr}, where ℱFM(E)={f∈C(E):|f(x)|≤1, |f(x)−f(y)|≤ρ(x,y), x,y∈E},〈f,μ〉=∫Ef(x)μ(dx) \matrix{{{{\cal F}_{FM}}\left(E \right) = \left\{{f \in C\left(E \right):\left| {f\left(x \right)} \right| \le 1,\,\,\,\left| {f\left(x \right) - f\left(y \right)} \right| \le \rho \left({x,y} \right),\,\,x,y \in E} \right\},} \cr {\langle f,\mu \rangle = \int_E {f\left(x \right)\mu \left({dx} \right)}} \cr}

As usual, by B(x, r) we denote the open ball in E centered at x and radius r > 0. For the fixed set A ⊂ E we define the indicator function 𝟙_A : E → {0, 1} as 𝟙_A(x) = 1 for x ∈ A and 𝟙_A(x) = 0 otherwise.

A function 𝒦: E × ℬ(E) → [0, 1] is called a (sub)stochastic kernel if for each A ∈ ℬ(E), x ↦ 𝒦(x, A) is a measurable map on E, and for each x ∈ E, A ↦ 𝒦(x, A) is a (sub)probability Borel measure on ℬ(E).

An operator P : ℳ₊(E) → ℳ₊(E) is called a Markov operator if:

• P (α₁μ₁ + α₂μ₂) = α₁Pμ₁ + α₂Pμ₂ for α₁, α₂ ∈ ℝ₊, μ₁, μ₂ ∈ ℳ₊(E),

• Pμ(E) = μ(E) for μ ∈ ℳ₊(E).

A Markov operator P is called regular if there is a linear operator U : B(E) → B(E), called the dual operator to P, such that (1) 〈f, Pμ〉=〈U f, μ〉for all f∈B(E), μ∈ℳ+(E). \matrix{{\left\langle {f,\,P\mu} \right\rangle = \left\langle {U\,f,\,\mu} \right\rangle} \hfill & {{\rm{for}}\,{\rm{all}}\,\,f \in B\left(E \right),\,\,\,\mu \in {{\cal M}_ +}\left(E \right)} \hfill \cr}. If P is a regular Markov operator then the function 𝒦: E × ℬ(E) → [0, 1], given by 𝒦(x, A) = Pδ_x(A) for x ∈ E, A ∈ ℬ(E), is a stochastic kernel. On the other hand, an arbitrary given stochastic kernel 𝒦 defines a regular Markov operator P and its dual operator U by formulas: Pμ(A)=∫EK(x,A)μ(dx)for μ∈ℳ+(E), A∈ ℬ(E) \matrix{{P\mu \left(A \right) = \int_E {{\cal K}\left({x,A} \right)\mu \left({dx} \right)}} \hfill & {{\rm{for}}\,\,\mu \in} \hfill \cr} {{\cal M}_ +}(E),\,\,\,A \in \,{\cal B}(E) and Uf(x)=∫Ef(y)K(x,dy)for x∈E, f∈B(E). Uf\left(x \right) = \int_E {\matrix{{f\left(y \right){\cal K}\left({x,dy} \right)} \hfill & {{\rm{for}}\,\,x \in E,\,\,\,f \in B\left(E \right)} \hfill \cr}}.

Let us note that the operator U can be extended to a linear operator defined on the space of all bounded below Borel functions B̅(E) so that (1) holds for all f ∈ B̅(E).

A regular Markov operator P is called Feller if U f ∈ C(E) for every f ∈ C(E).

We say that μ_∗ ∈ ℳ₊(E) is invariant with respect to P if Pμ_∗ = μ_∗.

If there exists an invariant measure μ_∗ ∈ ℳ₁(E) and a constant β ∈ [0, 1) such that, for every μ∈ℳ1V(E) \mu \in {\cal M}_1^V\left(E \right) and some constant C(μ) ∈ ℝ, we have ‖Pnμ−μ*‖FM≤C(μ)βnfor all n∈ℕ, \matrix{{{{\left\| {{P^n}\mu - {\mu_*}} \right\|}_{FM}} \le C\left(\mu \right){\beta^n}} \hfill & {{\rm{for}}\,{\rm{all}}\,\,n \in {\mathbb{N}}} \hfill \cr}, then μ_∗ is called exponentially attracting. If an operator P has such an exponentially attracting invariant probability measure, then the operator P is said to be exponentially ergodic.

It is well known that for every stochastic kernel 𝒦 and any fixed measure μ ∈ ℳ₁(E), we can always define on suitable probability space, say (Ω, ℱ, Prob_μ), a discrete-time homogeneous Markov chain {X_n}_n∈ℕ0 for which (2) Probμ(X0∈A)=μ(A)for A∈ℬ(E),K(x,A)=Probμ(Xn+1∈A|Xn=x)for x∈E, A∈ℬ(E), n∈ℕ0. \matrix{{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {\rm{Prob}}_{\mu}\left({{X_0} \in A} \right) = \mu \left(A \right)} \hfill & {{\rm{for}}\,A \in {\cal B}\left(E \right),} \hfill \cr {{\cal K}\left({x,A} \right) = {\rm{Pro}}{{\rm{b}}_\mu}({X_{n + 1}} \in A|{X_n} = x)} \hfill & {{\rm{for}}\,x \in E,\,A \in {\cal B}\left(E \right),\,n \in {{\mathbb{N}}_0}.}\cr} Then the Markov operator P corresponding to the kernel (2) describes the evolution of the distributions μ_n(·) := Prob_μ(X_n ∈ ·), that is μn+1=Pμnfor n∈ℕ0. \matrix{{{\mu_{n + 1}} = P{\mu_n}} \hfill & {{\rm{for}}\,\,n \in {{\mathbb{N}}_0}} \hfill \cr}. In our further considerations, we will use the symbol 𝔼_μ for the expectation with respect to Prob_μ. If μ = δ_x for some x ∈ E, we will write 𝔼_x.

We say that a time-homogeneous Markov chain evolving on the space E² (endowed with the product topology) is a Markovian coupling of some stochastic kernel 𝒦: E × ℬ(E) → [0, 1] whenever its stochastic kernel B : E² × ℬ(E²) → [0, 1] satisfies B(x,y,A×E)=K(x,A)andB(x,y,E×A)=K(y,A) \matrix{{B\left({x,y,A \times E} \right) = {\cal K}\left({x,A} \right)} \hfill & {{\rm{and}}} \hfill & {B\left({x,y,E \times A} \right) = {\cal K}\left({y,A} \right)} \hfill \cr} for all x, y ∈ E and A ∈ ℬ(E). Let us underline this, that if Q : E² × ℬ(E²) → [0, 1] is a substochastic kernel satisfying Q(x,y,A×E)≤K(x,A)andQ(x,y,E×A)≤K(y,A), \matrix{{Q\left({x,y,A \times E} \right) \le {\cal K}\left({x,A} \right)} \hfill & {{\rm{and}}} \hfill & {Q\left({x,y,E \times A} \right) \le {\cal K}\left({y,A} \right)} \hfill \cr}, for all x, y ∈ E and A ∈ ℬ(E), then we can always construct a Markovian coupling of 𝒦 whose transition function B satisfies Q ≤ B. For this purpose, it suffices to define the family {R(x, y, ·) : x, y ∈ E} of measures on ℬ(E²), which on A × B ∈ ℬ(E²) are given by R(x,y,A×B)=(K(x,A)−Q(x,y,A×E))(K(y,B)−Q(x,y,E×B))1−Q(x,y,E2) R\left({x,y,A \times B} \right) = {{\left({{\cal K}\left({x,A} \right) - Q\left({x,y,A \times E} \right)} \right)\left({{\cal K}\left({y,B} \right) - Q\left({x,y,E \times B} \right)} \right)} \over {1 - Q\left({x,y,{E^2}} \right)}} if Q(x, y, E²) < 1, and R(x, y, A × B) = 0 otherwise. Then B := Q + R is a stochastic kernel satisfying Q ≤ B, and that the Markov chain with transition function B is a coupling of 𝒦.

Let {X_n}_n∈ℕ0 be a Markov chain evolving in E with transition kernel 𝒦 and initial distribution μ. Suppose μ_∗∈ ℳ₁(E) is the unique invariant measure of the Markov operator P corresponding to the kernel given by (2). For any n ∈ ℕ and any Borel function g : E → ℝ define sn(g)=g(X1)+…+g(Xn)n,σ2(g)=limn→∞𝔼μ*(sn2(g)). \matrix{{{s_n}\left(g \right) = {{g\left({{X_1}} \right) + \ldots + g\left({{X_n}} \right)} \over {\sqrt n}},} \cr {{\sigma^2}\left(g \right) = \mathop {\lim}\limits_{n \to \infty} {{\mathbb{E}}_{{\mu_*}}}\left({s_n^2\left(g \right)} \right).} \cr} Denote by Φs_n(g) the distribution of s_n(g) and put ḡ = g − 〈g, μ_∗〉.

Let g : E → ℝ be a Borel function such that 〈g², μ_∗〉 < ∞. By definition, {g(X_n)}_n∈ℕ0 satisfies the CLT condition, if σ²(ḡ) < ∞ and Φ_sn(ḡ) converges weakly to 𝒩(0; σ²(ḡ)), as n → ∞.

Proving the CLT we will use the following theorem proved in [4].

Let (Y, ρ) be a Polish space and let I = {1, ..., N} for a fixed positive integer N. We define X : = Y × I and consider the space (X, ρ_c), where (3) ρc((y1,i), (y2,j))=ρ(y1,y2)+cΨ(i,j),(y1,i), (y2,j)∈X, \matrix{{{\rho_c}\left({\left({{y_1},i} \right),\,\left({{y_2},j} \right)} \right) = \rho \left({{y_1},{y_2}} \right) + c\Psi \left({i,j} \right),} \hfill & {\left({{y_1},i} \right),\,\left({{y_2},j} \right) \in X} \hfill \cr}, where (4) Ψ(i,j)={1 for i≠j,0 for i=j. \Psi \left({i,j} \right) = \left\{{\matrix{{1\,\,\,{\rm{for}}\,\,\,i \ne j,} \hfill \cr {0\,\,\,{\rm{for}}\,\,\,i = j.} \hfill \cr}} \right. The constant c > 0 will be defined later. Let Θ be a compact interval.

Our considerations are conducted for a discrete-time dynamical system related to the stochastic process {(Y (t), ξ(t))}_t∈ℝ+, which evolves through random jumps in the space X. We proceed to provide a description of this process.

Assume that we have a finite collection of maps, called semiflows, Π_i : ℝ₊ × Y → Y, i ∈ I, which are continuous with respect to each variable and satisfy for every i ∈ I and each y ∈ Y, the following conditions: Πi(0,y)=yandΠi(s+t,y)=Πi(s,Πi(t,y))for s,t∈ℝ+. \matrix{{{\Pi_i}\left({0,y} \right) = y} \hfill & {{\rm{and}}} \hfill & {{\Pi_i}\left({s + t,y} \right) = {\Pi_i}\left({s,{\Pi_i}\left({t,y} \right)} \right)} \hfill & {{\rm{for}}\,\,s,t \in {{\mathbb{R}}_ +}} \hfill \cr}.

The process {Y (t)}_t∈ℝ+ evolves between jumps according to one of the transformation Π_i, whose index i is determined by {ξ(t)}_t∈ℝ+.

Let π_ij : Y → [0, 1], i, j ∈ I be a matrix of continuous functions such that ∑j∈Iπij(y)=1for i∈I,y∈Y. \matrix{{\sum\limits_{j \in I} {{\pi_{ij}}\left(y \right) = 1}} \hfill & {{\rm{for}}\,i \in I,y \in Y} \hfill \cr}.

Just after every jump, the semiflow is changed from Π_i to Π_j in accordance with π_ij and at the moment of a jump the process {Y (t)}_t∈ℝ+ shifts to the new state by a function ω(θ, ·): Y → Y, which is randomly chosen from a given set {ω(θ, ·) : θ ∈ Θ}. We assume that Y × Θ ∋ (y, θ) ↦ ω(θ, y) ∈ Y is continuous and that the probability of choosing ω(θ, ·) is related with density functions Θ ∋ θ ↦ p(θ, y), y ∈ Y, such that (θ, y) ↦ p(θ, y) is continuous. Moreover, we assume that the intensity of jumps is given by a Lipschitz-continuous function λ : Y → (0, ∞), which satisfies the following conditions: (5) λ_=infy∈Yλ(y)>0andλ¯=supy∈Yλ(y)<∞. \matrix{{\underline \lambda = \mathop {\inf}\limits_{y \in Y} \lambda \left(y \right) > 0} \hfill & {{\rm{and}}} \hfill & {\overline \lambda = \mathop {\sup}\limits_{y \in Y} \lambda \left(y \right) < \infty} \hfill \cr}.

The evolution of {(Y (t), ξ(t))}_t∈ℝ+ can be described as follows. Assume that the process starts at some point (y₀, i₀) ∈ Y × I. Then Y(t)=Πi0(t,y0)andξ(t)=i0,0≤t<t1, \matrix{{Y\left(t \right) = {\Pi_{{i_0}}}\left({t,{y_0}} \right)} \hfill & {{\rm{and}}} \hfill & {\xi \left(t \right) = {i_0},} \hfill & {0 \le t < {t_1}} \hfill \cr}, where t₁ depends on y₀ and i₀. At time t₁ the process {Y (t)}_t∈ℝ+ jumps to the new position y1:=ω(θ1,Y(t1−))=ω(θ1,Πi0(t1,y0)), {y_1}: = \omega \left({{\theta_1},Y\left({{t_1} -} \right)} \right) = \omega \left({{\theta_1},{\Pi_{{i_0}}}\left({{t_1},{y_0}} \right)} \right), where θ₁ ∈ Θ is randomly chosen in accordance with the distribution given by density θ ↦ p(θ, Π_i0 (t₁, y₀)). In the next step, we choose randomly i₁ ∈ I such that the probability of choosing i₁ is given by π_i0i1 (y₁). Then Y(t)=Πi1(t−t1,y1)andξ(t)=i1,t1≤t<t2. \matrix{{Y\left(t \right) = {\Pi_{{i_1}}}\left({t - {t_1},{y_1}} \right)} \hfill & {{\rm{and}}} \hfill & {\xi \left(t \right) = {i_1},} \hfill & {{t_1} \le t < {t_2}} \hfill \cr}. We repeat the above steps with the point (y₁, i₁) instead of (y₀, i₀). The next steps are defined by induction.

Assuming that t₀ = 0, we get Y(t)=Πin(t−tn,yn)andξ(t)=in,tn≤t<tn+1,n∈ℕ0. \matrix{{Y\left(t \right) = {\Pi_{{i_n}}}\left({t - {t_n},{y_n}} \right)} \hfill & {{\rm{and}}} \hfill & {\xi \left(t \right) = {i_n},} \hfill & {{t_n} \le t < {t_{n + 1}},n \in {{\mathbb{N}}_0}} \hfill \cr}.

Let us emphasize that in this work we study only the sequence of random variables given by the post-jump locations of the process {(Y (t), ξ(t))}_t∈ℝ+, namely the process {(Y_n, ξ_n)_n∈ℕ0, where Y_n = Y (τ_n), ξ_n = ξ(τ_n) for n ∈ ℕ₀ and τ_n is a random variable describing the jump time t_n.

We can now give the formal description of the model. Let us set a probability space (Ω, ℱ, Prob_μ) and define {(Y_n, ξ_n)}_n∈ℕ0 as follows. Let (Y₀, ξ₀): Ω → X be a random variable with arbitrary and fixed distribution μ ∈ ℳ₁(X). Further, let us define by induction the sequences of random variables {τ_n}_n∈ℕ0, {ξ_n}_n∈ℕ, {η_n}_n∈ℕ and {Y_n}_n∈ℕ, which describe the sequences {t_n}_n∈ℕ, {i_n}_n∈ℕ, {θ_n}_n∈ℕ and {y_n}_n∈ℕ respectively, such that the following conditions are fulfilled:

• The sequence τ_n : Ω → [0, ∞), n ∈ ℕ₀, where τ₀ = 0, is strictly increasing such that τ_n → ∞ a.e., and the increments Δτ_n = τ_n − τ_n−1 are mutually independent and their conditional distributions are given by Probμ(Δτn+1≤t|Yn=y and ξn=i)=1−e−L(t,y,i)for t∈ℝ+, \matrix{{{\rm{Pro}}{{\rm{b}}_\mu}(\Delta {\tau_{n + 1}} \le t|{Y_n} = y\,\,{\rm{and}}\,\,{\xi_n} = i) = 1 - {e^{- L\left({t,y,i} \right)}}} \hfill & {{\rm{for}}\,t \in {{\mathbb{R}}_ +}} \hfill \cr}, where y ∈ Y, i ∈ I and L is given by (6) L(t,y,i)=∫0tλ(Πi(s,y))ds. L\left({t,y,i} \right) = \mathop \int \nolimits_0^t \lambda \left({{\Pi_i}\left({s,y} \right)} \right)ds.

• The sequence ξ_n : Ω → I, n ∈ ℕ satisfies the following condition Probμ(ξn=j|Yn=y, ξn−1=i)=πij(y)for i,j∈I,y∈Y. \matrix{{{\rm{Pro}}{{\rm{b}}_\mu}({\xi_n} = j|{Y_n} = y,\,\,{\xi_{n - 1}} = i) = {\pi_{ij}}\left(y \right)} \hfill & {{\rm{for}}\,i,j \in I,y \in Y} \hfill \cr}.

• η_n : Ω → Θ, n ∈ ℕ, is defined as follows Probμ(ηn+1∈D|Πξn(Δτn+1,Yn)=y)=∫Dp(θ,y)dθ {\rm{Pro}}{{\rm{b}}_\mu}({\eta_{n + 1}} \in D|{\Pi_{{\xi_n}}}\left({\Delta {\tau_{n + 1}},{Y_n}} \right) = y) = \int_D {p\left({\theta,y} \right)d\theta} for all D ∈ ℬ(Θ) and y ∈ Y.

• Y_n : Ω → Y, n ∈ ℕ, are given in following way Yn+1=ω(ηn+1,Πξn(Δτn+1,Yn))for n∈ℕ0. \matrix{{{Y_{n + 1}} = \omega \left({{\eta_{n + 1}},{\Pi_{{\xi_n}}}\left({\Delta {\tau_{n + 1}},{Y_n}} \right)} \right)} \hfill & {{\rm{for}}\,n \in {{\mathbb{N}}_{\rm{0}}}} \hfill \cr}.

U0=(Y0,ξ0),Uk=(Y0,τ1,…,τk,η1,…,ηk,ξ0,…,ξk)for k∈ℕ, \matrix{{{U_0} = \left({{Y_0},{\xi_0}} \right),} \hfill & {{U_k} = \left({{Y_0},{\tau_1}, \ldots,{\tau_k},{\eta_1}, \ldots,{\eta_k},{\xi_0}, \ldots,{\xi_k}} \right)} \hfill & {{\rm{for}}\,k \in {\mathbb{N}},} \hfill \cr}

We can easily check that the process {(Y_n, ξ_n)}_n∈ℕ0 is a time-homogeneous Markov chain with phase space X such that the evolution of the distributions μ_n(·) := Prob_μ((Y_n, ξ_n) ∈ ·) can be described by the Markov operator P : ℳ₊(E) → ℳ₊(E) given by (7) Pμ(A)=∑j∈I∫Θ∫X∫0∞𝟙A(ω(θ,Πi(t,y)),j)λ(Πi(t,y))e−L(t,y,i)×πij(ω(θ,Πi(t,y))p(θ,Πi(t,y)))dtdμ(dy,di)dθ P\mu \left(A \right) = \sum\limits_{j \in I} {\int_\Theta {\int_X {\int_0^\infty {{{\mathbb{1}}_A}\left({\omega \left({\theta,{\Pi_i}\left({t,y} \right)} \right),j} \right)\lambda \left({{\Pi_i}\left({t,y} \right)} \right){e^{- L\left({t,y,i} \right)}} \times {\pi_{ij}}\left({\omega \left({\theta,{\Pi_i}\left({t,y} \right)} \right)p\left({\theta,{\Pi_i}\left({t,y} \right)} \right)} \right)dtd\mu \left({dy,di} \right)d\theta}}}} for μ ∈ ℳ₊(E), A ∈ ℬ(E). Its dual operator U : B(E) → B(E) is given by Uf(y,i)=∑j∈I∫Θ∫0∞f(ω(θ,Πi(t,y)),j))λ(Πi(t,y))e−L(t,y,i)×πij(ω(θ,Πi(t,y)))p(θ,Πi(t,y))dtdθ Uf\left({y,i} \right) = \sum\limits_{j \in I} {\int_\Theta {\int_0^\infty {f(\omega (\theta,{\Pi_i}(t,y)),j))} \lambda ({\Pi_i}(t,y)){e^{- L(t,y,i)}} \times {\pi_{ij}}(\omega (\theta,{\Pi_i}(t,y)))p(\theta,{\Pi_i}(t,y))dtd\theta}} for x ∈ E, f ∈ B(E), where L is defined in (6).

We apply the following assumptions:

• (A1)

  There is y_∗ ∈ Y such that supy∈Y∫0∞eλ_t∫Θρ(ω(θ,Πi(t,y*)),y*)p(θ,Πi(t,y))dθdt<∞for i ∈ I. \matrix{{\mathop {\sup}\limits_{y \in Y} \int \nolimits_0^\infty {e^{\underline \lambda t}}\int_\Theta {\rho (\omega (\theta,{\Pi_i}(t,{y_*})),{y_*})p\left({\theta,{\Pi_i}\left({t,y} \right)} \right)d\theta dt < \infty}} \hfill & {{\rm{for}}\,i\, \in \,I.} \hfill \cr}

• (A2)

  There exist constants γ ∈ ℝ, L > 0, and a bounded on bounded subsets of Y function ℒ: Y → ℝ₊ such that the following inequality is satisfied ρ(Πi(t,y1),Πj(t,y2))≤Leγtρ(y1,y2)+tℒ(y2)Ψ(i,j) \rho \left({{\Pi_i}\left({t,{y_1}} \right),{\Pi_j}\left({t,{y_2}} \right)} \right) \le L{e^{\gamma t}}\rho \left({{y_1},{y_2}} \right) + t{\cal L}\left({{y_2}} \right)\Psi \left({i,j} \right) for t ∈ ℝ₊, y₁, y₂ ∈ Y, i, j ∈ I, where Ψ(i, j) is given by (4).

• (A3)

  There exists a constant M > 0 such that ∫Θρ(ω(θ,y1),ω(θ,y2))p(θ,y1)dθ≤Mρ(y1,y2)for y1,y2∈Y. \int_\Theta {\matrix{{\rho \left({\omega \left({\theta,{y_1}} \right),\omega \left({\theta,{y_2}} \right)} \right)p\left({\theta,{y_1}} \right)d\theta \le M\rho \left({{y_1},{y_2}} \right)} \hfill & {{\rm{for}}\,\,{y_1},{y_2} \in Y} \hfill \cr}.}

• (A4)

  There exists S > 0 such that |λ(y1)−λ(y2)|≤Sρ(y1,y2)for y1,y2∈Y. \matrix{{\left| {\lambda \left({{y_1}} \right) - \lambda \left({{y_2}} \right)} \right| \le S\rho ({y_1},{y_2})} \hfill & {{\rm{for}}\,{y_1},{y_2} \in Y} \hfill \cr}.

• (A5)

  There exists T > 0 and W > 0 such that ∑J∈I|πij(y1)−πij(y2)|≤Tρ(y1,y2)for y1,y2∈Y, i∈I,∫Θ|p(θ,y1)−p(θ,y2)|dθ≤Wρ(y1,y2)for y1,y2∈Y. \matrix{{\sum\limits_{J \in I} {\left| {{\pi_{ij}}\left({{y_1}} \right) - {\pi_{ij}}\left({{y_2}} \right)} \right| \le T\rho \left({{y_1},{y_2}} \right)}} \hfill & {{\rm{for}}\,\,{y_1},{y_2} \in Y,\,\,i \in I,} \hfill \cr {\int_\Theta {\left| {p\left({\theta,{y_1}} \right) - p\left({\theta,{y_2}} \right)} \right|d\theta \le W\rho \left({{y_1},{y_2}} \right)}} \hfill & {{\rm{for}}\,\,{y_1},{y_2} \in Y.} \hfill \cr}

• (A6)

  There exists ɛ_π > 0 and ɛ_p > 0 such that ∑J∈Imin{πi1j(y1),πi2j(y2)}≥επfor y1,y2∈Y, i1,i2∈I,∫Θ(y1,y2)min{p(θ,y1),p(θ,y2)}dθ≥εpfor y1,y2∈Y, \matrix{{\sum\limits_{J \in I} {\min \{{\pi_{{i_1}j}}\left({{y_1}} \right),{\pi_{{i_2}j}}\left({{y_2}} \right)\} \ge {\varepsilon_\pi}}} \hfill & {{\rm{for}}\,\,{y_1},{y_2} \in Y,\,\,{i_1},{i_2} \in I,} \hfill \cr {\int_{\Theta ({y_1},{y_2})} {\min \{p\left({\theta,{y_1}} \right),p\left({\theta,{y_2}} \right)\} d\theta \ge {\varepsilon_p}}} \hfill & {{\rm{for}}\,\,{y_1},{y_2} \in Y,} \hfill \cr} where Θ(y₁, y₂) = {θ ∈ Θ : ρ(ω(θ, y₁), ω(θ, y₂)) ≤ ρ(y₁, y₂)}.

As it was written at the beginning of this section, the constant c appearing in (3) is needed to be sufficiently large and depends on constants appearing in conditions (A1)–(A4). For more details, we refer to [6].

In [6] it is shown, that if the conditions (A1)–(A6) hold and the constants occurring in them satisfy the inequality (8) LMλ¯+γ<λ_, LM\overline \lambda + \gamma < \underline \lambda, then the operator P given by (7) is exponentially ergodic.

We would like to justify the conditions (A1)–(A6) stated above. The condition (A2) is met by a quite large class of semiflows defined on reflexive Banach spaces which can be generated by some differential equations engaging dissipative operators [10]. It happens often [5] then that the condition (A1) is a consequence of the conditions (A2) and (A3). The conditions (A3)–(A6) concerning assumptions such as contractivity are quite natural and in our setting commonly used regarding ergodic properties. One may consult [13, 15] for details.

In this section we prove the CLT for model {(Y_n, ξ_n)}_n∈ℕ0 described in the previous section. Our proof is strongly based on the proof of Theorem 4.1 in [4]. The proof will require extensions of the conditions (A1) and (A3) of the model, namely:

• (A1)′

  There exists y͂ > 0 such that supy∈Y∫0∞eλ_t∫Θ[ρ(ω(θ,Πi(t,y˜)),y˜)]2p(θ,Πi(t,y))dθdt<∞for i ∈ I. \matrix{{\mathop {\sup}\limits_{y \in Y} \mathop \int \nolimits_0^\infty {e^{\underline \lambda t}}\int_\Theta {{{[\rho (\omega (\theta,{\Pi_i}(t,\tilde y)),\tilde y)]}^2}p\left({\theta,{\Pi_i}\left({t,y} \right)} \right)d\theta dt < \infty}} \hfill & {{\rm{for}}\,i\, \in \,I.} \hfill \cr}

• (A3)′

  There exists M^′ > 0 such that ∫Θ[ρ(ω(θ,y1), ω(θ,y2))]2p(θ,y1)dθ≤M′[ρ(y1,y2)]2for y1,y2 ∈ Y. \matrix{{\int_\Theta {{{[\rho (\omega (\theta,{y_1}),\,\omega (\theta,{y_2}))]}^2}p\left({\theta,{y_1}} \right)d\theta \le M'{{[\rho ({y_1},{y_2})]}^2}}} \hfill & {{\rm{for}}\,{y_1},{y_2}\, \in \,Y.} \hfill \cr}

Let us also define a function V : X → ℝ₊ by (9) V(y,i)=ρ(y,y˜)for (y,i)∈X, \matrix{{V\left({y,i} \right) = \rho \left({y,\tilde y} \right)} \hfill & {{\rm{for}}\,\,\left({y,i} \right) \in X,} \hfill \cr} where y͂ is the constant found by the condition (A1)′.