Abstract
Fix an irrational number \(\alpha \), and consider a random walk on the circle in which at each step one moves to \(x+\alpha \) or \(x\alpha \) with probabilities 1/2, 1/2 provided the current position is x. If an observable is given we can study a process called an additive functional of this random walk. One can formulate certain relations between the regularity of the observable and the Diophantine properties of \(\alpha \) implying the central limit theorem. It is proven here that for every Liouville angle there exists a smooth observable such that the central limit theorem fails. We construct also a Liouville angle such that the central limit theorem fails with some analytic observable. For Diophantine angles some counterexample is given as well. An interesting question remained open.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Fix \(\alpha \in {\mathbb {R}}\), and consider a Markov process \((Y_n^\alpha )_{n\ge 1}\) defined on some probability space \((\Omega , \mathcal F, {\mathbb {P}})\) with the evolution governed by the transition kernel
whose initial distribution, i.e. the distribution of \(Y_1^\alpha \), is the Lebesgue measure (here \(\mathcal B ({\mathbb {S}}^1 )\) stands for the \(\sigma \)algebra of Borel subsets of \({\mathbb {S}}^1\)). One can easily verify the process is stationary. More work is needed to show the Lebesgue measure is the unique possible choice for the law of \(Y_1^\alpha \) to make the process stationary (see e.g. Theorem 7 and Remark 8 in [23]). In particular \((Y_n^\alpha )\) is ergodic, which means that if \(A\in \mathcal B ({\mathbb {S}}^1 )\) is such that \(p(x, A)=1\) for Lebesgue a.e. \(x\in {\mathbb {S}}^1\) then A is of the Lebesgue measure 0 or 1 (see e.g. Sect. 5 in [15], p. 37, for characterizations of ergodicity and the relation to the notion of ergodicity in dynamical systems).
This paper is devoted to the central limit theorem (CLT for short) for additive functionals of \((Y_n^\alpha )\), i.e. processes of the form \(\big (\varphi (Y^\alpha _1)+\cdots +\varphi (Y^\alpha _n)\big )\), where a function \(\varphi : {\mathbb {S}}^1 \rightarrow {\mathbb {R}}\) is usually called an observable. For convenience we assume that \(\int \varphi (x)dx=0\). We say that CLTholds for the process if
for some \(\sigma >0\). The validity of CLTdepends on Diophantine properties of \(\alpha \). An angle \(\alpha \) is called Diophantine of type \((c,\gamma )\), \(c>0\), \(\gamma \ge 2\) if
An angle \(\alpha \) is Liouville if it is not Diophantine of type \((c,\gamma )\) for any choice of \(c>0\), \(\gamma \ge 2\).
These and similar processes has been widely studied in the literature.

Kesten [16, 17] investigated the limit distribution of
$$\begin{aligned} D_N(x,\alpha )=\sum _{n=0}^{N1} \varphi (x+n\alpha )  N\int _{\mathbb {S}^1}\varphi (x)dx, \end{aligned}$$where \(\varphi \) is the characteristic function of some interval and \((x,\alpha )\) is uniformly distributed in \({\mathbb {S}}^1\times {\mathbb {S}}^1\). This was later generalized to higher dimensions by Dolgopyat and Fayad [9, 10].

Sinai and Ulcigrai [22] considered a similar problem when \(\varphi \) is nonintegrable meromorphic function.

In the above examples a point in the space is chosen randomly thus one calls it a spatial CLT. One can also fix a point in the space \(x\in {\mathbb {S}}^1\), an angle \(\alpha \) and, given N, pick randomly an integer number \(n\in [1, N]\). The question arise what is the limit distribution of \(D_n(x,\alpha )\) as N is growing. This kind of limit theorems are called temporal. The first limit theorem in this flavour was proven by Beck [1, 2]. For further development see e.g. [11, 5, 12].

Sinai [21] considered a situation where one draws \(+\alpha \) or \(\alpha \) with a probability distribution depending on the position in the circle (the method was to study a related random walk in random environment). He proved the unique ergodicity and stability of the process when \(\alpha \) is Diophantine. Recently Dolgopyat et al. [13] studied the behaviour in the Liouvillean case.

Borda [4] considered even a more general situation where several angles are given and one chooses one of them randomly. Given \(p\in (0,1]\), he formulated certain Diophantine conditions implying CLTfor all \(\varphi \) in the class of pHölder functions. Thus the author was concerned about what assumptions one should put on the angles of rotation to imply CLTfor all observables in a given class.
The situation here resembles the one from the last point, but here we rather touch the question how regular an observable should be to imply CLTif \(\alpha \) is given. Namely, using celebrated result by Kipnis and Varadhan [18] we prove the following statement.
Proposition 1
Let us assume \(\alpha \) to be Diophantine of type \((c,\gamma )\), \(\gamma \ge 2\). If a nonconstant function \(\varphi \in C^{r}\), \(r>\gamma 1/2\) (possibly \(r=\infty \)), is such that \(\int \varphi (x)dx=0\) then there exists \(\sigma >0\) such that
In particular, CLTholds if \(\alpha \) is Diophantine of an arbitrary type and \(\varphi \) is smooth.
The result is included for the sake of completeness, not because of novelty. This (or slightly different) statement has been proven independently by several people using various methods related to harmonic analysis [4, 8, Sect. 8], [24, Sect. 7.5], [25].
By Proposition 1CLT holds if \(\varphi \) is smooth and \(\alpha \) is Diophantine of an arbitrary type. It is natural to ask then if for every Liouville \(\alpha \) there exists a smooth \(\varphi \) for which CLTfails. It is also natural to ask if CLTfails if analytic observables are considered. This leads us to the following theorems showing dichotomy between the behaviour of Liouville and Diophantine random rotation, similar to the one appearing in smooth conjugacy results for circle diffeomorphisms (see the beginning of Chapter I.3 in [7]).
Theorem 1
There exists an irrational \(\alpha \) and \(\varphi \in C^\omega ({\mathbb {S}}^1)\) such that CLTfails.
Note that by Proposition 1 the angle in the assertion must be Liouville.
Theorem 2
Let \(\alpha \) be an irrational number. Let us assume there exist \(c>0\), \(\gamma >5\) such that
Let r be the largest positive integer with \(r<\frac{\gamma }{2}\frac{3}{2}\). Then there exist \(\varphi \in C^r\) such that CLTfails.
The only reason for making the assumption \(\gamma >5\) is to ensure \(\frac{\gamma }{2}\frac{3}{2}\) greater than 1, so that the condition \(r<\frac{\gamma }{2}\frac{3}{2}\) is satisfied for at least one positive integer r. A slightly changed proof of Theorem 2 yields the following.
Theorem 3
Let \(\alpha \) be Liouville. Then there exists \(\varphi \in C^\infty ({\mathbb {S}}^1)\) such that CLTfails.
Let us end this section with an interesting open problem. An angle \(\alpha \) is called badly approximable when it is Diophantine of type (c, 2) for some \(c>0\) (for instance, every quadratic irrational is badly approximable). Proposition 1 yields if \(\varphi \) is \(C^2\) then the additive functional satisfies CLT. Unfortunately, Theorem 2 does not give any counterexample in that case. This leads to a natural question: does CLTholds if \(\alpha \) is badly approximable (e.g. \(\alpha \) is the golden ratio) and \(\varphi \) is \(C^1\)?
2 The Poisson Equation and Central Limit Theorem
One of methods of proving CLTfor additive functionals of Markov chains is the GordinLifšic method [14], which is to be roughly explained in present section (note that in [4, 24, 25] different techniques have been used). Before that let us define the operator
where \(B({\mathbb {S}}^1)\) is the space of Borel measurable functions. By the very definition of a Markov process, if \((Y_n^\alpha )\) is defined on \((\Omega , \mathcal F, {\mathbb {P}})\) then
where p is the transition function (1).
Let \(\varphi : {\mathbb {S}}^1 \rightarrow {\mathbb {R}}\) be a square integrable function (with respect to the Lebesgue measure) with \(\int \varphi (x)dx=0\). To show the convergence of \(\frac{1}{\sqrt{n}}(\varphi (Y_1)+\cdots +\varphi (Y_n))\) to the normal distribution we solve so called Poisson equation^{Footnote 1}\(T\psi  \psi =\varphi \), where \(\psi \in L^2({\mathbb {S}}^1)\) is unknown. If the solution \(\psi \) exists then we can write
When divided by \(\sqrt{n}\), the second term tends to zero in probability. It is sufficient then to show CLTfor the first process, which is an ergodic, stationary martingale by (3). For such processess CLTis valid (see [6]). Thus the assertion follows provided the solution of the Poisson equation exists.
Observe that \((IT) u_n=(1\cos (2\pi n \alpha )) u_n\) for \(u_n(x)=\exp (2\pi i n x)\), \(x\in {\mathbb {S}}^1\), \(n\in {\mathbb {Z}}\). Therefore the trigonometric system \((u_n)_{n\in {\mathbb {Z}}}\) is also the orthonormal system of eigenvectors of \(IT\) with corresponding eigenvalues \(1\cos (2\pi n \alpha )\), \(n\in {\mathbb {Z}}\). We deduce the nth Fourier coefficient of \((IT)\psi \), \(\psi \in L^2({\mathbb {S}}^1)\), is of the form \((1\cos (2\pi n \alpha ))\hat{\psi }(n)\), \(n\in {\mathbb {Z}}\). This yields a recipe to find \(\psi \) when \(\varphi \) is given. Namely, \(\psi \) should be a square integrable function whose Fourier series coefficient are
while \(\hat{\psi }(0)\) is an arbitrary real number. Note we use here also the assumption that \(\hat{\varphi }(0)=\int \varphi (x)dx=0\). Indeed, \(1\cos (0)=0\) implies that we must have \(\hat{\varphi }(0)=0\) to solve the equation. What remains to do is to show the convergence
to make sure the object with Fourier coefficients (5) is indeed a square integrable function.
In fact the solution of the Poisson equation does not have to exists to have CLT. Note the processes under consideration are reversible, which means that the distribution of random vectors \((Y^\alpha _1, \ldots , Y^\alpha _n)\) and \((Y^\alpha _n, \ldots , Y^\alpha _1)\) are the same for every natural n or, equivalently, the operator T is selfadjoint. In celebrated paper [18] (see Theorem 1.3 therein) the authors have proven the condition \(\varphi \in \text {Im}(IT)\) can be relaxed to
where \(\sqrt{IT}\) is the square root of \(IT\) (recall the square root of a positive semidefinite, selfadjoint operator P acting on a Hilbert space is the operator \(\sqrt{P}\) with the property \((\sqrt{P})^2=P\)). Since the nth Fourier coefficient of the function \((IT)\psi \), \(\psi \in L^2({\mathbb {S}}^1)\) is given by \((1\cos (2\pi n \alpha ))\hat{\psi }(n)\), we easily deduce that \(\sqrt{IT}\) is well defined on \(L^2({\mathbb {S}}^1)\) and the nth Fourier coefficient of the function \(\sqrt{IT}\psi \), \(\psi \in L^2({\mathbb {S}}^1)\), is given by \(\sqrt{1\cos (2\pi n \alpha )}\hat{\psi }(n)\). Thus (7) leads to the condition
weaker than (6). Moreover, [18] (see (1.1) therein) delivers a formula for \(\sigma \), which reads here as
Clearly, \(\sigma ^2<\infty \) if (8) is satisfied and \(\sigma >0\) if \(\varphi \) is nonconstant.
We are in position to prove Proposition 1. We recall the statement for the convenience of the reader.
Proposition 1
Let us assume \(\alpha \) to be Diophantine of type \((c,\gamma )\), \(\gamma \ge 2\). If a nonconstant function \(\varphi \in C^{r}\), \(r>\gamma 1/2\) (possibly \(r=\infty \)), is such that \(\int \varphi (x)dx=0\) then there exists \(\sigma >0\) such that
In particular, CLTholds if \(\alpha \) is Diophantine of an arbitrary type and \(\varphi \) is smooth.
Proof
We are going to prove (8) is satisfied. Fix \(\alpha \) and \(\varphi \) as above. Clearly \(\sum _{n\in {\mathbb {Z}}} \hat{\varphi }(n)^2<\infty \) since \(\varphi \) is square integrable, therefore the problem is when \(\cos (2\pi \alpha n)\) is close to 1, which happens exactly when \(\alpha n\) is close to some integer. To handle this we will use the fact that \(\alpha \) is Diophantine of type \((c, \gamma )\). This means
By Taylor’s formula \(\cos (2 \pi (p+x))1=\frac{(2\pi x)^2}{2}+o(x^2)\) for \(p\in {\mathbb {Z}}\). As a consequence there exists \(\eta >0\) such that
for an arbitrary \(n \in {\mathbb {Z}}\). If \(\varphi \in C^r\) then \(\hat{\varphi }(n)\le Cn^{r}\) for some constant C thus
for every n. It is immediate that if \(r>\gamma \frac{1}{2}\), then the series (8) is convergent. This implies CLTby Theorem 1.3 in [18].
Clearly, if \(\varphi \) is a trigonometric polynomial, then series (8) becomes a finite sum and thus the condition is trivially satisfied. This yields another proposition, which will be used in the proof of Theorem 2.\(\square \)
Proposition 2
Let us assume \(\alpha \) to be irrational. If \(\varphi \) is a nonconstant trigonometric polynomial with \(\int \varphi (x)dx=0\) then there exists \(\sigma >0\) such that
3 Auxiliary Results
In the proofs three lemmas will be pivotal. Given integer \(q\ge 1\), \(\eta \in (0,1/2)\), define \(G_q^\eta \) to be the subset of \({\mathbb {S}}^1\) containing all points whose distance from the set \(\{ 0, \frac{1}{q}, \ldots , \frac{ q1}{q} \}\) (where \(\cos (2\pi q x)\) attains value 1) is less than \(\frac{\eta }{q}\). Clearly \(\text {Leb}(G_q^\eta )=2\eta \) whatever q is. Recall that \((Y^{\alpha }_n)\) stands for the Markov process defined on some probability space \((\Omega , \mathcal F, {\mathbb {P}})\) with transition function (1) and \(Y_1^\alpha \sim \text {Leb}\).
Lemma 1
Let \(\alpha =\frac{p}{q}\), \(\varphi (x)=2^{q} \cos (2 \pi q x)\) and let \(s\in (0,1)\). Let N be an arbitrary natural number with \(2^{q1}N^{1s}>2\). If \(\alpha '\) is sufficiently close to \(\alpha \) then
Note the assertion is more difficult to obtain when s is close to 1.
Proof
The result is the consequence of the invariance of \(\varphi \) under the action of the rotation of angle \(\alpha \). In particular the set \(G_q^\eta \) is invariant for every \(\eta >0\). Take N like in the statement, and choose \(\alpha '\) so close to \(\alpha \) that \(x+n\alpha ' \in G_q^{1/6}\) for \(n\le N\) and \(x\in G_q^{1/12}\). By the definition of \(G_q^\eta \), the value of \(\varphi \) on \(G_q^{1/6}\) is greater or equal to \(2^{q}\cos (2\pi /6)\ge 2^{q}\cdot 1/2\). Thus \(\varphi (x+n\alpha ')\ge 2^{q}\cdot 1/2=2^{q1}\) for \(n\le N\) and \(x\in G_q^{1/12}\). This yields
Using the facts that \(Y_1^{\alpha '}\sim \text {Leb}\), \(\text {Leb}(G_q^{1/12})=1/6\) and \(2^{q1}N^{1s}>2\) we have
which yields the assertion. \(\square \)
A slightly different lemma is the following.
Lemma 2
Let \(\alpha \) be an irrational number, \(s\in (1/2, 1)\), \(c>0\), \(\gamma \ge 2\). If \(\alpha \) satisfies
for some pair of integers p, q, \(q\not = 0\), then
where \(\varphi (x)=q^{(\gamma 1)(1s)}\cos (2\pi q x)\), \(N=\lfloor \frac{q^{\gamma 1}}{16c}\rfloor \).
Proof
If \(\alpha  \frac{p}{q}\le \frac{c}{q^\gamma }\) and \(k\le \frac{q^{\gamma 1}}{16c}\) then
Thus \(z+n\alpha \in G^{1/8}_q\) for all \(z\in G^{1/16}_q\) and integers n with \(n\le N\). On the other hand, the value of \(\varphi \) on \(G^{1/8}_q\) is greater or equal to \(q^{(\gamma 1)(1s)}\cos (\frac{2 \pi }{8})=\frac{\sqrt{2}}{2}\cdot q^{(\gamma 1)(1s)}\). By the same reasoning as in the proof of Lemma 1 we have
and consequently
\(\square \)
Take \(\alpha =p/q\) rational (p/q is in the irreducible form) and the corresponding process \((Y_n^\alpha )\). If the initial point \(Y_1^\alpha \) is already known, then we know also each \(Y^\alpha _n\), \(n\in {\mathbb {N}}\), is contained almost surely in the orbit of \(Y_1^\alpha \) under the action of the rotation of angle \(\alpha \), \(\{Y_1^\alpha , Y_1^\alpha +\alpha , \ldots , Y_1^\alpha +(q1)\alpha \}\) (this set is finite, since \(\alpha \) is rational). The process \((Y^\alpha _n)\) can be therefore treated as a finite state Markov chain.
If q is odd, then the process \((Y^\alpha _n)\) treated as a finite state Markov chain is aperiodic and irreducible. Its stationary distribution the uniform distribution on the set \(\{Y_1^\alpha , Y_1^\alpha +\alpha , \ldots , Y_1^\alpha +(q1)\alpha \}\) (every state is of measure 1/q). It follows from Theorem 8.9 (p. 131) [3] that
where the constants A and \(\rho \in (0,1)\) are independent of x (since neither the space nor the transition probabilities depend on x). Let \(\varphi (x)=a\cos (2\pi q' x)\) for some \(a>0\) and \(q'\) not a multiplicity of q. Since p/q is assumed to be in an irreducible form, \(p/q\cdot q'\) is not an integer and thus we have
for every \(x\in {\mathbb {S}}^1\), which is equivalent to say that the integral of \(\varphi \) with respect to the stationary distribution of \((Y_n^\alpha )\) (treated as a finite state Markov chain) equals zero. Moreover, using (11) gives
for \(n\ge 1\). Thus
The next lemma is essentially the consequence of the central limit theorem for finite state irreducible and aperiodic Markov chains. However, using (12) we may deduce it in simpler way.\(\square \)
Lemma 3
Let \(\alpha =\frac{p}{q}\) be rational (in irreducible form), qodd. Let \(\varphi (x)=a\cos (2\pi q' x)\) for some \(a>0\) and \(q'\) not a multiplicity of q. If \(s>1/2\) then for every \(\varepsilon >0\) and \(\delta >0\) there exists N such that
for \(n \ge N\).
Proof
It follows from the Chebyshev inequality. We have
By (12) and the stationarity of the process each of the numerators in the sum does not exceed \(\frac{2A q \Vert \varphi \Vert ^2_\infty }{1\rho }\), thus the second term is bounded by
The entire expression tends to zero since \(s>1/2\). The assertion follows. \(\square \)
4 Proof of Theorem 1
Fix an arbitrary \(s\in (\frac{1}{2}, 1)\). We are going to construct an angle \(\alpha \) and an observable \(\varphi \) with \(\int \varphi (x)dx=0\) such that there exist infinitely many n’s with
Consequently the process does not satisfy CLTsince CLTwould imply the above quantity tends to zero. First we shall define inductively a sequence of numbers \(\alpha _k\) convergent to some \(\alpha \) along with certain observables \(\varphi _k\). Then we will put \(\varphi =\sum _k \varphi _k\) and use some relations between \(\alpha _k\) and \(\varphi _k\) established during the induction process to get the above assertion.
Put \(\alpha _1=\frac{1}{3}=\frac{p_1}{q_1}\) (when we represent a rational number as a fraction of integers we always assume it to be in an irreducible form, so here \(p_1=1\) and \(q_1=3\)), and set \(\varphi _1(x)=2^{q_1} \cos (2\pi q_1 x)\). Take \(N_1\) so large that \(2^{q_11}N_1^{1s}>2\) and apply Lemma 1 to obtain an angle \(\alpha _2=\frac{p_2}{q_2}\), with \(q_2>q_1\) and \(q_2\) odd, such that
Define \(\varphi _2(x)=2^{q_2}\cos (2\pi q_2 x)\). Take \(N_2>N_1\) so large that \(2^{q_21} N_2^{1s}>2\). Clearly \(q_1\) is not a multiplicity of \(q_2\), hence by Lemma 3 we can assume that \(N_2\) is so large that
Again use Lemma 1 to obtain an angle \(\alpha _3=\frac{p_3}{q_3}\), with \(q_3>q_2\) and \(q_3\) odd, such that
We assume also the number \(\alpha _3\) is so close to \(\alpha _2\) that (13) and (14) still hold with \(\alpha _2\) replaced by \(\alpha _3\). This combined with (15) gives
and
Assume \(\alpha _k=\frac{p_k}{q_k}\), \(N_i\), \(\varphi _i\) are already defined, \(k\ge 3\), \(i<k\). These objects satisfy the relations
and
Define \(\varphi _k(x)=2^{q_k} \cos (2 \pi q_k x)\) and take \(N_k>N_{k1}\) so large that \(2^{q_k1} N_k^{1s}>2\) and
for \(i=1,\ldots , k1\), by Lemma 3. Use Lemma 1 to get a number \(\alpha _{k+1}=\frac{p_{k+1}}{q_{k+1}}\), with \(q_{k+1}>q_k\), \(q_{k+1}\) odd, such that
We should take care that \(\alpha _{k+1}\) is so close to \(\alpha _k\) that (16), (17) and (18) still hold with \(\alpha _k\) replaced by \(\alpha _{k+1}\). With this modification, (16) and (18) become
while (17) and (19) can be rewritten as
This completes the induction. Observe there is no inconsistency in assuming that \(q_{k+1}\)’s grow so fast that
This way the sequences of numbers \((\alpha _k)\), \((N_k)\) and functions \((\varphi _k)\) are defined. Set \(\alpha =\lim _{k\rightarrow \infty } \alpha _k\) and \(\varphi =\sum _{k=1}^\infty \varphi _k\). When passing to the limit, inequality (20) becomes
while (21) yields
The function \(\varphi \) is analytic. Indeed, by design
where \(c_k= \Vert \varphi _j\Vert _\infty =2^{q_j}\) if \(k=q_j\) and zero otherwise. Thus the Fourier coefficients of \(\varphi \) decay exponentially fast, which implies \(\varphi \) to be analytic.^{Footnote 2} Obviously \(\int \varphi (x)dx=0\) by the Lebesgue convergence theorem. Observe also that (22) combined with \(\Vert \varphi _i\Vert _\infty =2^{q_i}\) yield
We are in position to complete the proof, i.e. to show that
for every k. To this end fix k and write
From (25) it easily follows that the absolute value of the second summand on the righthand side is less than \(\frac{1}{2}\) almost surely. Thus
By (23) and (24) it follows that
which is the desired assertion.
5 Proof of Theorems 2 and 3
The entire section is devoted to the proof of Theorem 2. In the end we will give a short remark how to change the proof to get Theorem 3.
Fix an irrational \(\alpha \) and numbers \(c>0\), \(\gamma \ge 2\) such that
for infinitely many pairs \(p,q \in {\mathbb {Z}}\), \(q\not = 0\). Take r to be the largest possible integer with \(r<\frac{\gamma }{2}\frac{3}{2}\). The function \(s\longmapsto (\gamma 1)(1s)1\) is decreasing, \(s\in [\frac{1}{2}, 1)\), and its value at \(s=\frac{1}{2}\) is \(\frac{\gamma }{2}\frac{3}{2}\), thus by continuity we can choose \(s>\frac{1}{2}\) such that \(r<(\gamma 1)(1s)1\). For this choice of s we are going to construct an observable \(\varphi \) with \(\int \varphi (x)dx=0\) such that
for infinitely many n’s. Consequently CLTis violated.
Take arbitrary \(p_1, q_1 \in {\mathbb {Z}}\), \(q_1\not = 0\), satisfying (26). Set \(\varphi _1(x) = q_1^{(\gamma 1)(1s)}\cos (2 \pi q_1 x)\) and apply Lemma 2 to get
where \(N_1=\lfloor \frac{q_1^{\gamma 1}}{16c}\rfloor \). By Proposition 2 the additive functional \((\varphi _1(Y^\alpha _1)+\cdots +\varphi _1(Y^\alpha _n))\) satisfies CLT, thus for N sufficiently large
Let us take \(p_2, q_2\in {\mathbb {Z}}\), \(q_2\not = 0\), such that (26) holds, \(N_2=\lfloor \frac{q_2^{\gamma 1}}{16c}\rfloor \) satisfies (28) and
(this will imply that the inequality (27) is not affected too much when \(\varphi _1\) replaced by \(\varphi _1+\varphi _2\)). Lemma 2 yields
Assume \(N_k\), \(p_k\), \(q_k\) are already defined. Let us choose a pair \(q_{k+1}, p_{k+1} \in {\mathbb {Z}}\) with (26), where \(q_{k+1}>q_k\) is so large that
Moreover, using Lemma 3 we demand that \(q_{k+1}\) is so large that
where \(N_{k+1}=\lfloor \frac{q_{k+1}^{\gamma 1}}{16c}\rfloor \). Finally we use Lemma 2 to get
where \(\varphi _{k+1}(x)=q_{k+1}^{(\gamma 1)(1s)}\cos (2\pi q_{k+1} x)\). When the induction is complete put
By assumption \(r<(\gamma 1)(1s)1\), therefore we can take \(\varepsilon >0\) so that \(r=(\gamma 1)(1s)(1+\varepsilon )\). If one differentiates this series r times, then it still converges uniformly (with the rate at least \(q^{(1+\varepsilon )}\)). Therefore Theorem 7.17 (p. 152) in [19] yields \(\varphi \) is \(C^r\).
Now it remains to show that
for every \(k\in {\mathbb {N}}\). We proceed analogously to the proof of Theorem 1. Fix k. We have
The application of (30) yields the second term is bounded by \(\frac{\sqrt{2}}{8\cdot (16 c)^{1s}}\) a.s. Therefore
The application of (31) and (32) yields Theorem 2.
To demonstrate Theorem 3 observe that for \(\alpha \) Liouville there exist sequences of integers \(p_k\), \(q_k\) with
The only difference with the proof of Theorem 2 is that p, q are chosen from this sequence. Then again \(\varphi =\sum _m \varphi _m\), and the series is uniformly convergent after differentiating it rtimes for an arbitrary r. This implies \(\varphi \in C^\infty \). The rest remains unchanged.
Notes
In dynamical systems theory this equation (with T replaced by a Koopman operator) is called a cohomological equation. The name “Poisson equation” is more common in theory of stochastic processes, probably due to the fact that writing down the corresponding equation for a Brownian motion, which is a continuous time Markov process, gives \(\frac{1}{2} \Delta \varphi = \psi \), where \(\Delta \) is the Laplace operator. Note \(\frac{1}{2} \Delta \) is the infinitesimal generator of the Brownian motion.
Indeed, \(\varphi \) is defined as a series on the circle, however by the exponential convergence it can be extended to some neighbourhood of the unit disc \({\mathbb {D}}\) in the complex plane \(\mathbb {C}\). Then \(\varphi \) becomes a sum of holomorphic functions convergent uniformly on compact subsets of the domain of \(\varphi \). Theorem 10.28 (p. 214) in [20] implies \(\varphi \) is holomorphic.
References
Beck, J.: Randomness of the square root of 2 and the Giant Leap, part 1. Period. Math. Hung. 60(2), 137–242 (2010)
Beck, J.: Randomness of the square root of 2 and the Giant Leap, part 2. Period. Math. Hung. 62(2), 127–246 (2011)
Billingsley, P.: Probability and measure. In: Wiley Series in Probability and Mathematical Statistics, 3rd edn. Wiley, New York (1995)
Borda, B.: Equidistribution of random walks on compact groups II. The Wasserstein metric. Bernoulli 27(4), 2598–2623 (2021)
Bromberg, M., Ulcigrai, C.: A temporal central limit theorem for realvalued cocycles over rotations. Ann. Inst. Henri Poincaré Probab. Stat. 54(4), 2304–2334 (2018)
Brown, B.M.: Martingale central limit theorems. Ann. Math. Stat. 42, 59–66 (1971)
de Melo, W., van Strien, S.: OneDimensional Dynamics. Springer, Berlin (1993)
Derriennic, Y., Lin, M.: The central limit theorem for Markov chains with normal transition operators, started at a point. Probab. Theory Relat. Fields 119, 508–528 (2001)
Dolgopyat, D., Fayad, B.: Deviations of ergodic sums for toral translations I. Convex bodies. Geom. Funct. Anal. 24(1), 85–115 (2014)
Dolgopyat, D., Fayad, B.: Deviations of ergodic sums for toral translations II. Boxes. Publ. Math. Inst. Hautes Études Sci. 132, 293–352 (2020)
Dolgopyat, D., Sarig, O.: Temporal distributional limit theorems for dynamical systems. J. Stat. Phys. 166(3–4), 680–713 (2017)
Dolgopyat, D., Sarig, O.: Quenched and annealed temporal limit theorems for circle rotations. Astérisque (415, Quelques aspects de la théorie des systèmes dynamiques: un hommage à JeanChristophe Yoccoz. I), 59–85 (2020)
Dolgopyat, D., Fayad, B., Saprykina, M.: Erratic behavior for 1dimensional random walks in a Liouville quasiperiodic environment. Electron. J. Probab. 26 (2021)
Gordin, M.I., Lifšic, B.A.: Central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR 239(4), 766–767 (1978)
Hairer, M.: Ergodic properties of Markov processes. Lecture notes (2006). http://www.hairer.org/notes/Markov.pdf
Kesten, H.: Uniform distribution \({\rm mod}\,1\). Ann. Math. 2(71), 445–471 (1960)
Kesten, H.: Uniform distribution \({\rm mod}\ 1\). II. Acta Arith 7, 355–380 (1961/1962)
Kipnis, C., Varadhan, S.R.S.: Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104(1), 1–19 (1986)
Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. McGrawHill Book Co, New York (1976)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGrawHill Book Co, New York (1987)
Sinai, Y.: Simple random walks on Tori. J. Stat. Phys. 94(3–4), 695–708 (1999)
Sinai, Y.G., Ulcigrai, C.: A limit theorem for Birkhoff sums of nonintegrable functions over rotations. In: Geometric and Probabilistic Structures in Dynamics. Contemporary Mathematics, vol. 469, pp. 317–340. American Mathematical Society, Providence (2008)
Szarek, T., Zdunik, A.: Stability of iterated function systems on the circle. Bull. Lond. Math. Soc. 48(2), 365–378 (2016)
Weber, M.: Dynamical Systems and Processes. IRMA Lectures in Mathematics and Theoretical Physics, vol. 14. European Mathematical Society, Zürich (2009)
Zdunik, A.: Personal communication
Acknowledgements
This research was supported by the Polish National Science Centre grant Preludium UMO2019/35/N/ST1/02363. I am grateful to Anna Zdunik for fruitful discussions and for sharing her notes with the proof of CLT. I would also like to thank Corinna Ulcigrai for providing references [5, 22]. I am grateful to two anonymous referees for many comments that helped improve the manuscript and for providing me reference [24]. Finally, I would like to thank Michael Lin for discussions and for pointing out that [18] applies here.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Marco Lenci.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Czudek, K. Some Counterexamples to the Central Limit Theorem for Random Rotations. J Stat Phys 189, 11 (2022). https://doi.org/10.1007/s10955022029757
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10955022029757