Discrete Holomorphic Local Dynamical Systems

214 Tien-Cuong Dinh and Nessim Sibony
converges in law to the normal distribution of mean 0 and variance σ. In particular,
ν-almost surely
1
logN
N
∑
n=1
1
n 1� n −1/2 ∑ n−1
i=0 Zi≤t0
�
� 1 t0 − t2
→ √2πσ e 2σ
−∞
2 dt,
for any t0 ∈ R. In the central limit theorem, we only get the ν-measure of the set
{N−1/2 ∑ N−1
n=0 Zn < t0} when N goes to infinity. Here, we get an information at
ν-almost every point for the logarithmic averages.
The almost sure central limit theorem can be deduced from the so-called almost
sure invariance principle (ASIP for short). In the case of i.i.d. random variables as
above, this principle compares the variables �ZN with Brownian motions and gives
some information about the fluctuations of �ZN around 0.
Theorem 1.90. Let (X,F ,ν) be a probability space. Let (Zn) be a sequence of i.i.d.
random variables with mean 0 and variance σ > 0. Assume that there is an α > 0
such that Zn is in L 2+α (ν). Then, there is another probability space (X ′ ,F ′ ,ν ′ )
with a sequence of random variables S ′ N on X ′ which has the same joint distribution
as SN := ∑ N−1
n=0 Zn, and a Brownian motion B of variance σ on X ′ such that
|S ′ N − B(N)|≤cN1/2−δ ,
for some positive constants c,δ. It follows that
|N −1/2 S ′ N − B(1)|≤cN −δ .
For weakly dependent variables, this type of result is a consequence of a theorem
due to Philipp-Stout [PS]. It gives conditions which imply that the ASIP holds.
Lacey-Philipp proved in [LP] that the ASIP implies the almost sure central limit
theorem. For holomorphic endomorphisms of P k , we have the following result due
to Dupont which holds in particular for Hölder continuous observables [DP2].
Theorem 1.91. Let f be an endomorphism of algebraic degree d ≥ 2 as above and
μ its equilibrium measure. Let ϕ be an observable with values in R ∪{−∞}
such that e ϕ is Hölder continuous, H := {ϕ = −∞} is an analytic set and
|ϕ| � |logdist(·,H)| ρ near H for some ρ > 0.If〈μ,ϕ〉 = 0 and ϕ is not a coboundary,
then the almost sure invariance principle holds for ϕ. In particular, the almost
sure central limit theorem holds for such observables.
The ASIP in the above setting says that there is a probability space (X ′ ,F ′ ,ν ′ )
with a sequence of random variables S ′ N on X which has the same joint distribution
as SN := ∑ N−1
n=0 ϕ ◦ f n , and a Brownian motion B of variance σ on X ′ such that
for some positive constants c,δ.
|S ′ N − B(N)|≤cN 1/2−δ ,