Spectra of stationary processes on Z
Mikhail Sodin (Tel Aviv University)
joint work with
Alexander Borichev (Aix-Marseille U.),
Benjamin Weiss (Hebrew U. of Jerusalem)
arXiv: January 2017
Bedlewo, May 2017
Finitely-valued stationary processes on Z
¯
ξ : Z → C stationary square integrable
process,
E ξ(n)ξ(m)
its
¯
covariance. By stationarity, E ξ(n)ξ(m)
= r (m − n).
The sequence (r (m))m∈Z is positive definite. Hence,
Z
t −m dρ(t) = ρb(m), ρ ∈ M+ (T).
r (m) =
T
ρ the spectral measure of ξ, spt(ρ) the spectrum of ξ
Question: What can be said about the spectral measures of
finitely-valued stationary sequences?
Main example: {0, 1}-stationary sequences:
(X , B, µ, T ) probability space with a measure preserving
automorphism T , Y ∈ B, 0 < µ(Y ) < 1.
Take x ∈ X , consider its orbit (T n x)n∈Z , and put ξ(n) = 1
whenever T n x ∈ Y and ξ(n) = 0 otherwise.
The main result
Theorem: Suppose the sequence ξ is finitely-valued and
spt(ρ) 6= T. Then ξ is periodic.
Question: What about stationary processes on Zd with d > 2?
A note on the terminology:
We say that the stationary process ξ on Z is periodic if there exists
N ∈ N s.t., almost every realization (ξ(n))n∈Z is N-periodic.
In this case, the covariance r (n) is also N-periodic and therefore
spt(ρ) ⊂ {t : t N = 1}.
We say that the stationary process ξ on Z has periodic realizations
if for almost every realization (ξ(n))n∈Z there exists N ∈ N s.t. this
realization is N-periodic. In this case, N can be a random number.
• If the process is ergodic, then these definitions are equivalent.
First approach: via spectra of realizations
ξ : Z → C wide stationary: E|ξ(n)|2 < ∞;
¯ + m)] do not depend on n.
E[ξ(n)] and E[ξ(n)ξ(n
E|ξ(n)|2 < ∞ & Borel-Cantelli
⇒ ∀α >
1
2
a.s. |ξ(n)| = o(|n|α ) as n → ∞.
⇒ ξ defines a random functional Fξ :
Fξ (ϕ) =
X
ξ(n)ϕ(−n),
b
ϕ ∈ C ∞ (T).
n∈Z
σ(ξ)
= spt(Fξ ): complement to the largest open set O ⊂ T s.t.
Fξ = 0.
O
• σ(ξ) is invariant w.r.t. translations of ξ (hence, is non-random
when ξ is ergodic);
• σ(ξ) is invariant w.r.t. the flip ξ(n) 7→ ξ(−n).
Spectra of individual realizations
Theorem 1: Suppose ξ : Z → C is a wide-stationary process with
zero mean, ρ is the spectral measure of ξ. Then
(A) a.s., σ(ξ) ⊆ spt(ρ);
(B) if O ⊂ T is an open set s.t. a.s. O ∩ σ(ξ) = ∅, then
O ∩ spt(ρ) = ∅.
The proof is a straightforward application of the linear isomorphism
between spanL2 (P) {ξ(n) : n ∈ Z} and L2 (ρ) (ξ(n) 7→ t n , n ∈ Z).
Corollary: Suppose ξ : Z → C is stationary, square integrable,
ergodic process. Then, a.s., σ(ξ) = spt(ρ).
Remark: Sometimes, it is more convenient to use other equivalent
definitions of the spectrum of the realization (ξ(n))n∈Z (Carleman,
Beurling)
Helson-type argument
Theorem 2: Suppose X ⊂ C is a finite set. Then any sequence
ξ : Z → X with σ(ξ) 6= T is N-periodic with N depending only on
X and σ(ξ).
Theorems 1 and 2 yield:
Theorem 3: Suppose X ⊂ C is a finite set, and ξ : Z → X is a
wide-stationary process with the spectral measure ρ. Suppose
spt(ρ) 6= T. Then the process ξ is periodic.
Remark: Helson proved Theorem 2 with N depending on the
sequence ξ, his proof uses a compactness argument. Our proof is
based on the following lemma.
δ-prediction
Lemma: Given positive δ, p, M and an open arc J ⊂ T, J¯ 6= T,
there exist n ∈ N and q0 , . . . qn−1 ∈ C s.t., for any sequence
C with kξkp 6
ξ : Z → P
M and σ(ξ) ⊂ J, the following holds
ξ(n) + n−1 qk ξ(k) < δ.
k=0
P |ξ(n)| 2
.
Here, kξk2p = n∈Z 1+|n|
p
To deduce Theorem 2, first, note that, since the sequence ξ attains
only finitely many values, knowing ξ(n) with a small error δ, we
know it precisely, so the lemma yields that, in assumptions of
Theorem 2, the values ξ(0), . . . , ξ(n − 1) uniquely determine ξ(n).
Repeating this process, we see that ξ(0), . . . , ξ(n − 1) uniquely
determine the whole sequence ξ.
Then, by the pigeonhole principle, the sequence ξ must be
periodic.
2
Historical digression:
Szegő used a somewhat similar idea to prove a theorem about
Taylor series with coefficients attaining a finite set of values:
Theorem (Szegő): Suppose that a sequence of Taylor coefficients
(fn )n>0 attainsPfinitely many values and the sum of the Taylor
series f (z) = n>0 fn z n can be analytically continued through an
arc of the unit circle. Then the sequence (fn ) is eventually periodic
and f is a rational function with poles at the roots of unity.
The second approach: via Szegő’s-type polynomial
approximation
X ⊂ C uniformly discrete set: δX = inf x6=y ∈X |x − y | > 0.
en (ρ) = distL2 (ρ) 1l, Pn0 , ρ ∈ M+ (T)
Here, Pn0 are algebraic polynomials of degree n vanishing at the
origin.
P
2
Spectral condition (Θ):
n>1 en (ρ) < ∞.
Theorem 4: Suppose X ⊂ C is a uniformly discrete set, ξ : Z → X
stationary process with the spectral measure satisfying condition
(Θ). Then a.e. realization (ξ(n))n∈Z is periodic.
Corollary: Suppose that in the assumptions of Theorem 4 the
process ξ is ergodic. Then ξ periodic.
Sketch of the proof of Theorem 4
By the linear isomorphism between spanL2 (P) {ξ(n) : n ∈ Z} and
L2 (ρ) (ξ(n) 7→ t n ), given N ∈ N there exist q0 , . . . , qN−1 ∈ C, s.t.
2 P 1
= eN (ρ)2 .
E ξ(N) + N
k=0 qk ξ(k)
1 P 1
⇒ P ξ(N) + N
k=0 qk ξ(k) > 2 δX 6
⇒ with probability > 1 −
8
δX2
4
δX2
eN (ρ)2
2
n>N en (ρ) ,
P
the whole realization
(ξ(n))n∈Z is determined by ξ(0), . . . , ξ(N − 1).
(Θ) ⇒ the probability space Ω is countable.
Note that, by stationarity of ξ, there exists a measure preserving
transformation τ of (Ω, P).
S
⇒ Ω = j Ωj , each Ωj consists of finitely many atoms of equal
weights and is τ -invariant.
⇒ τ permutes elements of Ωj , that is, ∀j, τ Ω is periodic.
2
j
Condition (Θ)
en (ρ) = distL2 (ρ) 1l, Pn0 , ρ ∈ M+ (T)
P
2
Spectral condition (Θ):
n>1 en (ρ) < ∞.
Szegő’s theorem: en (ρ) → 0 ⇔
0
T log |ρ | dm
R
= −∞
Remark: spt(ρ) 6= T ⇒ en (ρ) 6 Ce −cn .
Question: when (en (ρ))n∈N has a power decay?
Locally, a power decay of (en (ρ))n∈N is is governed the
exponentially deep zeroes in ρ:
Theorem 5: Let ρ ∈ M+ (T) and let β be a positive parameter.
Rπ
(A) Suppose that −π e β/|θ| dρ(e iθ ) < ∞. Then en (ρ) . n−C β .
(B) Suppose dρ & e −β/|θ| dm. Then en (ρ) & n−β/(2π) .
P.S. The McMillan condition
¯
r (m) = E[ξ(0)ξ(m)]
the covariance sequence of a stationary
process on Z that attains values from the set X ⊂ C.
P
(Aij )16i,j6n is such that, for every x1 , . . . , xn ∈ X , i,j Aij xi x̄j > 0.
Then,
X
X
¯
Aij r (j − i) = E
Aij ξ(i)ξ(j)
> 0.
i,j
i,j
This can be restated in terms of the spectral measure ρ:
Z X
Aij t i−j dρ(t) > 0.
T
i,j
In the case X = {±1} this condition is also the sufficient one
(McMillan stated this without proof, the proof was published later
by Shepp).
The End