A short note on one-dependent
trigonometric determinantal probability
measures 1
Donato Michele Cifarelli and Sandra Fortini
Università Bocconi, Milano.
Abbreviated title: Trigonometric determinantal probabilities.
Abstract
We provide a characterization of one-dependent determinantal probability measures, based on predictive distributions. The characterization is applied to give a new proof that one-dependent determinantal
probability measures are two-block-factors.
1. Introduction. We will start by recalling the definition of m-dependent
determinantal probability measure (see Broman 2005). Let f : [0, 1] → [0, 1]
be a Lebesgue-measurable function and let P f be the stationary probability
measure on the Borel sets of {0, 1}∞ defined on the cylinder sets by
P f {x ∈ {0, 1}∞ : xi1 = 1, . . . , xik = 1} := det[fˆ(ir − is )]1≤r,s≤k ,
where i1 , . . . , ik are distinct elements in N and k ≥ 1. Here fˆ denote the
1
AMS 2000 subject classification: 60G10
Keywords and phrases. Determinantal processes, k-dependance, k-block-factors.
1
Fourier coefficients of f , defined by
fˆ(k) :=
Z
1
f (x)e−i2πkx dx
(k ≥ 0).
0
In Lyons (2003), it is proved that P f is indeed a probability measure. In
fact, they showed this for the more general case of f : Td → [0, 1], where
Td := Rd /Zd . In this case, the resulting process is indexed by Zd . The
proof rests very strongly on the results in Lyons and Steif (2003).
If the function f : [0, 1] → [0, 1] is of the form
f (x) =
m
X
ak e−i2πkx ,
(1)
k=−m
then P f is an m-dependent probability measure, according to the definition
below.
Definition 1.1. Let (Xi )i∈N be a stochastic process with probability distribution P . The process (Xi )i∈N is called m-dependent if (Xi )i<k is independent of (Xi )i≥k+m for all integers k.
A probability measure P f , with f satisfying (1) is called m-dependent
trigonometric determinantal probability measure. These probability measures are special cases of general determinantal probability measures (see
Soshnikov, 2000, Lyons, 2003, Shirai and Takahashi, 2003, and the references
therein). Determinantal processes arise in numerous contexts as mathematical physics, random matrix theory, representation theory, ergodic theory, to
name a few. For a survey see Soshnikov (2000), for further results see Lyons
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(2003) and Shirai and Takahashi (2003). For results concerning the discrete
stationary case see Lyons and Steif (2003). In Lyons and Steif (2003), it is
asked whether P f is an (m + 1)-block-factor whenever f satisfies (1). We
D
recall the definition of m-block-factor. Let = denote equality in distribution.
Definition 1.2. A process (Xi )i∈N with probability distribution P is an
m-block-factor of an i.i.d. sequence if there exists a function h of m variables
D
and an i.i.d. process (Yi )i∈N such that (Xi )i∈N = (h(Yi , . . . , Yi+m−1 ))i∈N .
Every (m + 1)-block-factor is trivially m-dependent. The converse is
not true, in general (see Aaronson et al., 1989, Aaronson et al., 1992, Burton et al., 1993). De Valk (1993) used Hilbert space techniques to investigate the connections between one-dependent processes and two-block-factors
processes. The question whether m-dependent trigonometric determinantal
probability measures are (m+1)-block-factors is still open, as far as we know.
Broman (2005) has answered the question positively for m = 1, by finding a
Borel subset A of [0, 1]2 such that (1A (Yi , Yi+1 ))i∈N has probability law P f ,
when (Yi )i∈N are i.i.d. random variables with uniform distribution on [0, 1].
In Section 2, we will provide a simpler Borel set and an easier proof that
the probability distribution of the process is P f . Such proof is based on a
predictive characterization of one-dependent trigonometric determinanantal
probability measures (Theorem 2.1).
2. Main result. Let P f be a one-dependent determinantal probability
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measure, corresponding to
f (x) = a + bei2πx + be−i2πx ,
(2)
with a ∈ [0, 1] and |b| ≤ min(a/2, (1 − a)/2)). Since the process is onedependent and stationary, its probability distribution is uniquely determined
by the probabilities of the cylinders
{x ∈ {0, 1}∞ : x1 = . . . = xk = 1},
as k varies over the positive integers.
Define, for every (e1 , . . . , ek ) ∈ {0, 1}k , and k ≥ 1,
∆k (e1 , . . . , ek ) : = P f {x ∈ {0, 1}∞ : x1 = e1 , . . . , xk = ek }
= det[aij (e1 , . . . , ek )]1≤i,j≤k ,
where




aei (1 − a)1−ei




aij ((e1 , . . . , ek ) =
(−1)1−ei b






 0
if j = i
if j = i − 1, i + 1
otherwise.
The following is a characterization of one-dependent determinantal probability measures based on predictive distributions.
Theorem 2.1. Let (Xi )i∈N be a {0, 1}-valued stationary and one-dependent
process with probability distribution P . A necessary and sufficient condition
for P = P f with f as in (2) is



 P (X = 1) = a
1
(3)


 Cov(Xn+1 , Xn+2 |X1 , . . . , Xn ) = −b2
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P − a.s.
(n ≥ 0).
Proof.
Necessity. For every n ≥ 0, it holds



 ∆ (1) = a
1


 ∆n+2 (e1 , . . . , en , 1, 1) = a∆n+1 (e1 , . . . , en , 1) − b2 ∆n (e1 , . . . , en )
Hence (Xi )i∈N satisfies (3), when P = P f .
Sufficiency. The sequence (∆k (1, . . . , 1))k∈N is uniquely determined by




∆1 (1) = a




∆2 (1, 1) = a2 − b2






 ∆n+2 (1, . . . , 1) = a∆n+1 (1, . . . , 1) − b2 ∆n (1, . . . , 1) n ≥ 1.
On the other hand, since (Xi )i∈N is one-dependent, it holds, P -a.s.,
P (Xn+1 = 1, Xn+2 = 1|X1 = 1, . . . , Xn = 1)
= Cov(Xn+1 Xn+2 |X1 = 1, . . . , Xn = 1) + aE(Xn+1 |X1 = 1, . . . , Xn = 1).
It follows from (3) that




P (X1 = 1) = a




P (X1 = 1, X2 = 1) = a2 − b2






 P (X1 = 1, . . . , Xn+2 = 1) = aP (X1 = 1, . . . , Xn+1 = 1)−b2 P (X1 = 1, . . . , Xn = 1)
Hence P (X1 = 1, . . . , Xk = 1) = ∆k (1, . . . , 1) for every k ≥ 1. This proves
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the theorem.
(1)
(1)
Incidentally, we observe that ∆k (1, . . . , 1) = bk Ck (a/(2b)), where Ck
is the k-th ultraspherical polynomial of order 1.
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(n > 0).
Theorem 2.1 can be exploited to construct a two-block-factor with law
P f . Without loss of generality, we can suppose that 0 < a ≤ 1/2. Indeed, if
(Xi )i∈N is distributed according to P f and f satisfies (2), then (1 − Xi )i∈N
has probability law P g , with
g(x) = 1 − a − be−i2πx − bei2πx
(see Shirai and Takahashi 2003).
Let (Yi )i∈N be a sequence of i.i.d. random variables with uniform distribution on [0, 1]. We will prove that (1A (Yi , Yi+1 ))i∈N satisfies (3) for a
suitable choice of A, for example when A is the set
[0, d] × [0, 1/2] ∪ [1 − d, 1] × [0, 1/2] ∪ [0, 1] × [1/2, 1/2 + c],
with d = a/2 +
(4)
p
p
a2 /4 − b2 and c = a/2 − a2 /4 − b2 (see Figure 1).
Lemma 2.2. For every i ≥ 1, let Xi = 1A (Yi , Yi+1 ), where A is a Borel
subset of [0, 1]2 . If



 E(X1 ) = a
(5)


 E(Xn+1 (Xn+2 − a)|Yn+1 ) = −b2
(n ≥ 0),
then (Xi )i∈N satisfies (3).
Proof. From (5) and the properties of conditional expectation, it follows
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that
E(Xn+1 (Xn+2 − a)|X1 , . . . , Xn )
= E(E(Xn+1 (Xn+2 − a)|Y1 , . . . , Yn+1 )|X1 , . . . , Xn )
= E(E(Xn+1 (Xn+2 − a)|Yn+1 )|X1 , . . . , Xn )
= −b2
hold a.s., for every n ≥ 0. This completes the proof.
2
Based on Lemma 2.1, to construct a two-block-factor with law P f , it is
sufficient to find a Borel set A of [0, 1]2 that satisfies
(λ × λ)(A) = a
(6)
and
Z
λ(Ax )dx = aλ(As ) − b2
λ − a.s.,
(7)
As
where As = {x ∈ [0, 1] : (s, x) ∈ A} and λ is the Lebesgue measure on [0, 1].
We restrict ourselves to sets A that are symmetric with respect to {(x, y) ∈
[0, 1]2 : x = 1/2} and that satisfy (6). Then (7) holds if the following conditions are verified
for every s, t ∈ [0, 1/2], either As = At , or As ⊂ At and At \ As = [0, 1/2],
or At ⊂ As and As \ At = [0, 1/2];
there exists s in [0, 1/2] such that
R
As
λ(Ax )dx = aλ(As ) − b2 .
It is not difficult to find a set satisfying these conditions. The set (4) is
just an example.
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References
Aaronson, J., Gilat, D. and Keane, M. (1992) On the structure of 1dependent Markov chains. J. Theoret. Probab. 5 545-561.
Aaronson, J., Gilat, D., Keane, M. and de Valk, V. (1989) An algebraic
construction of a class of one-dependent processes. Ann. Probab. 17
128-143.
Broman, E. I. (2005) One-dependent trigonometric determinantal processes
are two-block-factors. Ann. Probab. 33 601-609.
Burton, R. M., Goulet, M. and Meester, R. (1993) On one dependent
processes and k-block-factors. Ann. Probab. 21 2157-2168.
Lyons, S. (2003) Determinantal probability measures. Publ. Math. Inst.
Hautes Études Sci. 98 167-212.
Lyons, S. and Steif, J. (2003) Stationary determinantal processes: Phase
transitions, Bernoullicity, entropy and domination. Duke Math. J.
120 515-575.
Shirai, T. and Takahashi, Y. (2003) Random point fields associated with
certain Fredholm determinants II: fermion shifts and their ergodic and
Gibbs properties. Ann. Probab. 31 1533-1564.
Soshnikov, A. (2000) Determinantal random point fields. Uspekhi Mat.
Nauk. 55 107-160.
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Valk, V. de (1993) Hilbert space representations of m-dependent processes.
Ann. Probab. 21 1550-1570.
Istituto di Metodi Quantitativi,
Viale Isonzo, 25, 20135 Milano, Italy.
E-mail: [email protected]
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