Ergodicity and central limit theorem for random interval homeomorphisms

i

√_n

i

^Σn


[0,1]


P_n(di)μ_∗(dy) for t ∈ R

it

exp

Φ_n(t) =
ϕ(f ¹(y)) + ·· · + ϕ(fⁿ(y))

and

Φ_n(t) =
x
ϕ(f ¹(x)) + ·· · + ϕ(fⁿ(x))

i

√_n

i

exp

it

P_n(di) for t ∈ R.
From the first part of the proof we know that there exists σ ≥ 0 such that for
every t ∈ R

Φ_n(t) → exp
_−t2_σ2
as n → ∞.

2
Therefore, to finish the proof it is sufficient to show for every t ∈ R the conver- gence

x
|Φ_n(t) − Φ_n(t)|→ 0 as n → ∞.
Since |e^itx1 − e^itx2 |≤ |t||x₁ − x₂| for every t ∈ R, for sufficiently large n ∈ N
and x ∈ [ε_n, 1 − ε_n] using μ_∗ ∈ P_M,α we have

| |_x
t
|Φ (t) − Φ_n(t)| ≤√ ·



(ϕ(f^j(x)) − ϕ(f^j(y))) P_n(di)μ_∗(dy)

n _n
|t|
[0,1]
i i
^Σn j=1

Σ

i

i

n

n

[ε ,1−ε ]
≤√
(ϕ(f^j(x)) − ϕ(f^j(y))) P_n(di)μ_∗(dy)

n
|t|
^{n n} j=1


α

+ ^√_n · 2n∗ϕ∗Mε_n

n
|t|

n

[ε ,1−ε ]

A \D

i

i
≤√
(ϕ(f^j (x))−ϕ(f ^j(y))) P_n(di)μ_∗(dy)

n
|t|
n n n


α
j=1
|t|

+ ^√_n · 2n∗ϕ∗Mε_n + ^√_n · 2n∗ϕ∗(P_n(Σ_n \ A_n)+ P_n(D_n)).
Since ε_n → 0 as n → ∞, for every x ∈ (0, 1) and t ∈ R, by (5.2) and using the
estimates P_n(B_n) ≤ (1 − β) √^n⊗ and P (D ) ≤ 2Mγ , we finally obtain
8

n
n n

lim
n→∞
|Φ_n(t) − Φ_n

20. 
    
    |≤
    lim
    

n→∞
2|t|

n

√

x
n

3
(C₁Ln ⁸ + n

∗ϕ∗(1 − β) √8 ^n⊗)

This completes the proof.
+ lim
n→∞

2|t|∗ϕ∗^√n((1 − δ) √4 ^n⊗ + Mε^α) = 0.

Corollary 2: If the assumptions of Theorem 4 are satisfied, then the sta-

n

k=0
tionary sequence ( ^{√1 [nt]} ϕ(X_k)) converges in distribution to σW (t), where
W (t), 0 ≤ t ≤ 1 is a Brownian motion. Moreover, for μ_∗-a.e. x ∈ (0, 1) the

n

k=0

k
sequence ( ^{√1 [nt]} ϕ(X^x)) also converges in distribution to σW (t), 0 ≤ t ≤ 1.

_{— n}

∞
Proof. From the proof of Theorem 4 it follows that the Maxwell–Woodroofe condition

n =1
n
3

j=1
U^jϕ < ∞


^L2(μ∗ ⁾

is satisfied. Applying Theorem 1.1 in [28] we obtain the first assertion. On the other hand, the second assertion follows from Theorem 2.7 in [9].


Acknowledgments. The authors wish to express their gratitude to an anony- mous referee for thorough reading of the manuscript and valuable remarks.

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