Ergodicity and central limit theorem for random interval homeomorphisms

The proof of the Central Limit Theorem

Let (f₁,..., f_N ; p₁,..., p_N ) be an admissible iterated function system and let P be the corresponding Markov operator. Let (X_n) be the Markov chain corre- sponding to P . This part of the paper is devoted to the proof of the quenched central limit theorem for the random process (ϕ(X_n)), where ϕ is a Lipschitz
function. The question of the quenched central limit theorem was raised, for re- versible Markov chains, by C. Kipnis and S. R. S. Varadhan in [20]. M. I. Gordin
and B. A. Lifˇsic proved that if μ_∗ is an ergodic invariant measure for a Markov operator P and ϕ is an L₂(μ_∗) coboundary, i.e., ϕ ∈ (I − U )L₂(μ_∗), then the quenched central limit theorem holds for μ_∗-a.e. x (see [8]). It is worth men- tioning here that in our case the support of μ_∗ could be a nontrivial subset of the interval [0, 1]. In fact, for some admissible iterated function systems the
unique invariant measure μ_∗ is distributed on a Cantor set (see [4]). Recently, it was proved that reversible chains satisfy a quenched invariance principle for


ϕ ∈ (I − U )L_q(μ_∗) ∩ L_p(μ_∗),
where 1 ≤ q ≤ 2 and p = q/(q − 1) (see [7]). Since ϕ in our considerations is a bounded function, to prove the quenched central limit theorem it is enough to check that ϕ ∈ (I − U )L_q(μ_∗) for some 1 ≤ q ≤ 2. Unfortunately, we have failed to do it. Indeed, we are unable to verify that

∞
Uⁿϕ(x)
n=1

is convergent μ_∗-a.s. for ϕ such that μ_∗, ϕ = 0. This holds true when the Markov operator P satisfies the spectral gap property in the Wasserstein–

Kantorovich metric. Although our system is asymptotically stable on (0, 1), its rate of convergence has not yet been determined. Our proof is therefore based on the Maxwell–Woodroofe theorem (see [27]). In fact, this theorem allows us
to prove the annealed central limit theorem. On the other hand, to prove the quenched central limit theorem for every x ∈ (0, 1) we construct some coupling between trajectories and evaluate the distance between Fourier transforms for
Markov processes starting at different initial points.
We are now in a position to formulate and prove the main results of our paper.

Theorem 4 (Central Limit Theorem): Let (f₁,... , f_N ; p₁,... , p_N ) be an ad- missible iterated function system, and let (X_n) be the corresponding stationary

Markov chain with the initial distribution μ_∗. If ϕ : [0, 1] → R is a Lipschitz function with _[0,1] ϕdμ_∗ = 0, then the random process (ϕ(X_n)) satisfies the central limit theorem, i.e., the limit

√
₂ ϕ(X₀)+ ·· · + ϕ(X_n) ²

exists and
σ := lim E
^n→∞ n

·· ·_{0 n}
ϕ(X )+ + ϕ(X )
√_n ⇒ N (0, σ) as n → ∞,
where ⇒ denotes convergence in distribution. Moreover, the same is true for
the process (ϕ(X^x)), where (X^x) is the corresponding Markov chain starting
n n
from an arbitrary point x ∈ (0, 1).
Proof. Since μ_∗ is ergodic for P by the uniqueness in Theorem 1, the chain is ergodic. Therefore to prove the first part of our theorem it is sufficient to show
the following condition (see Theorem 1 in [27]):

n



₃

∞


(5.1)


n=1
_n− ₂
j=1

2
U^jϕ

^L2(μ∗ ⁾
< ∞,

where ∗· ∗_L2_{(μ∗ )} denotes the L -norm with respect to the invariant measure μ_∗.
Let α, δ, ε, M be the constants given in Lemma 1 and let a be such that

6

0
M = ε^−α < ^a−α . Choose n ∈ N according to Lemma 4. Put J = [a, 1 − a] and

define

i

n

n
E := {i ∈ Σ : f √4 ^n⊗([ε
, 1 − ε_n
]) ⊂ J} for n ≥ n₀.

From Lemma 4 we have


P(E_n


1
) ≥ ₅

for n ≥ n₀.

Let Ω ⊆ Σ be the set given in Lemma 3 and define
A = {ij_|n : i ∈ E_n, j ∈ Ω}⊆ Σ_n.

√
It is easily seen that P_n(A) ≥ β > 0 for a certain β independent of n. We apply
Lemma 5 to the set A, k = ≤ ⁸ n for n ≥ n₀ and obtain some sequence of sets
A_n ⊆ Σ_n with P_n(Σ_n \ A_n) ≤ (1 − δ) √^n⊗ such that if i ∈ A , then i = i ··· i
8

n
1 k

and for every m = 1,...,k at least one of the sequences i_m, σi_m,..., σ^k−1i_m is dominated by A. Hence for i_m, m = 1,...,k and for x, y ∈ [ε_n, 1 − ε_n] we have

√



^|im^{|
j j} 4
^√₈ q
_√4
q

j=1
|f_i (x) − f_i (y)|≤ ≤
n + r + ≤
n + _{1 − q} ≤ 2 ≤
n + r + _{1 − q} ,

where r is the constant given in Lemma 3. Further, set

i

i
D_n := {i ∈ A_n : ∃_m≥ ^√4 _n⊗f^m(x) < ε_n or f^m(y) > 1 − ε_n}
and notice that for any sequence i ∈ A_n \ D_n and every x, y ∈ [ε_n, 1 − ε_n] we have

n

j

j

4

q

|f_i (x) − f_i (y)|≤ k

≤

n + r + _{1 − q}

3

(5.2)

≤ C₁n ⁸
√

j=1

for some positive constant C₁. Furthermore, P_n(D_n) ≤ 2Mγ_n, by Lemma 2.

Let B_n := Σ_n \ A_n. From Lemma 5 we obtain that P_n(B_n) ≤ (1 − β) √^n⊗.
8



∗ ∗
Denote by L the Lipschitz constant of ϕ. Since


ϕdμ = U^jϕdμ = 0
[0,1] [0,1]
for j ∈ N, we have

n

2

n





U^jϕ(x) μ_∗(dx)=

(U^jϕ(x)−U^jϕ(y))μ_∗(dy) μ_∗(dx)

2



j

^≤ [0,1]
[0,1]
n


j=1
|U ϕ(x)−U
2

j
ϕ(y)|
μ_∗(dx)μ_∗(dy).

From the fact that μ_∗([0, x]) ≤ M x^α and μ_∗([1−x, 1]) ≤ M x^α for every x ∈ [0, 1] we obtain the following inequality:

[0,1]
[0,1]


n





j
j=1


|U ϕ(x) − U

2

j
ϕ(y)|


μ_∗(dx)μ_∗(dy)

^{≤ [ε}n ^,1−εn^]
[εn ^,1−εn ^]
n




j=1
|U ϕ(x) − U
2


j
ϕ(y)|
μ_∗(dx)μ_∗(dy)

n

j
+ 16ε^αn²M∗ϕ∗².

Since ε = (1 −δ) √^n⊗, the last sequence is bounded by, say, C

2
1 4



> 0. However,

n


n

by estimate (5.2) for x, y ∈ [ε_n, 1 − ε_n] we have
j j ^{j j}

|U
j=1
ϕ(x) − U
ϕ(y)|≤
^An ^\Dn _j=1
|ϕ(f_i (x)) − ϕ(f_i (y))|dP_n(i)

j j
+ |ϕ(f (x)) − ϕ(f (y))|dP_n(i)
i i

n

^Dn j=1
+ |ϕ(f (x)) − ϕ(f (y))|dP_n(i)

j j
i i
^Bn j=1
3
≤LC₁n ⁸ + 2nLP_n(D_n)+ 2nLP_n(B_n).
By the fact that P_n(B_n) ≤ (1 − β) √^n⊗ and P (D ) ≤ 2Mγ , the last two
8

n
n n
sequences above are bounded and therefore there exists a positive constant C₃
such that

3

8

j=1
|U^jϕ(x) − U^jϕ(y)|≤ C n 3
for x, y ∈ [ε
, 1 − ε ].

n


n
Combining the above estimates we finally obtain
ⁿ 2
₂ 6

j

j
[0,1]
[0,1]
|U


j=1
ϕ(x) − U
ϕ(y)|
μ_∗(dx)μ_∗(dy) ≤ C₃ n ⁸ + C₂.

Therefore there exists a positive constant C such that

n
j=1
U^jϕ
^L2(μ∗ ⁾
3
≤ Cn ⁸ .

This proves (5.1). Application of Theorem 1 in [27] finishes the proof for the stationary sequence (X_n).

n
To derive the central limit theorem for (X^x), x ∈ (0, 1), observe that by conditions (2.3) and (2.4) the characteristic functions of the processes
^{1 1} x x
^√_n (ϕ(X₀)+ ··· + ϕ(X_n)) and ^√_n (ϕ(X₀ )+ ··· + ϕ(X_n ))
are of the form, respectively,