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Sankhy¯ a : The Indian Journal of Statistics 1999, Volume 61, Series A, Pt. 1, pp. 72-88

SOME NEW EXAMPLES OF PAIRWISE INDEPENDENT RANDOM VARIABLES By ANDRZEJ KÃLOPOTOWSKI Universit´e Paris XIII, Villetaneuse and JAMES B. ROBERTSON University of California, Santa Barbara SUMMARY. We give some new examples of pairwise independent random variables. Special attention is paid to the problem of the existence of a dynamical system with simple Lebesgue spectrum.

1.

Introduction

This paper is devoted to the study of sequences of pairwise independent random variables. Such sequences have some of the properties of the mutually independent ones such as the Borel–Cantelli law and the strong law of large numbers, but some others fail, like for example the central limit theorem or Kolmogorov’s zero–one law. It is not surprising because, contrary to the common opinion that the mutual independence is a very “natural” property, it is an extremely particular case in the wider class of pairwise independent random variables. Some explanations and indications about the real interest of this class are given in Robertson (1985, 1988). A general description of this class is not known. There are only a few methods of construction of such sequences (see Bretagnolle et al. (1995), Cuesta et al. (1991) and Robertson (1988a) for history and bibliography). Our very elementary research (Derriennic et al. 1988, 1991a,b) shows that the pairwise independence (as well as the exchangeability) is not constructive in the sense that one cannot build the corresponding finite dimensional laws by induction. Paper received. March 1997; revised May 1998. AMS (1980) subject classification. Primary 60A05, 60G10; secondary 47A35, 28D20. Key words and phrases. Stationary process, pairwise independence, m–dependence, Markov chain, dynamical system, simple Lebesgue spectrum.

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In this paper we only consider the stationary sequences of binary random variables (the non–stationary case being rather hopeless). In Section 2 we give all necessary definitions. In the next section we present Robertson and Womack’s finitary approach in a slightly generalized form given by Derriennic (1991). In Section 4 one can find a discussion of Bretagnolle’s example (Bretagnolle et al. 1995) and we show that its nature is also finitary. Section 5 contains some new examples which are variations on the theme of the preceeding one, but they have essentially different properties; for example we believe that Example 6 is not finitary. In the last section we discuss all these examples from the point of view of the famous problem (attributed to Stefan Banach) of the existence of a dynamical system with simple Lebesgue spectrum. It is very probable that such a system does not exist and our examples rather confirm this possibility. 2.

Definitions

This paper is addressed rather to the probabilists, but we hope that some ergodicians could find here something interesting, so we recall some primary notions of both of these domains. Here we exclusively consider strictly stationary sequences of two–valued (binary) random variables Xn , n ∈ Z, (Z denotes the set of integers) defined on some common probability space (Ω, F, P). A stationary sequence of random variables {Xn ; n ∈ Z} is called Rademacher sequence if P[Xn = +1] = P[Xn = −1] = 1/2, n ∈ Z. A Rademacher sequence is pairwise independent if and only if for each k > 1, P[X1 = Xk = +1] = 1/4 (which is equivalent to their orthogonality in the Hilbert space L2 (Ω, F, P)). It is m–dependent (m ≥ 1 fixed) if and only if for each k, l ∈ N and for each choice of values xi = +1; 1 ≤ i ≤ k, k + m + 1 ≤ i ≤ k + m + l, P[X1 = x1 , . . . , Xk = xk , Xk+m+1 = xk+m+1 , . . . , Xk+m+l = xk+m+l ] = P[X1 = x1 , . . . , Xk = xk ] · P[Xk+m+1 = xk+m+1 , . . . , Xk+m+l = xk+m+l ]. Sometimes we shall consider binary variables with values different from ±1; the corresponding definitions are analogous. Given any stationary sequence of binary random variables Xn , n ∈ Z, we can construct a probability measure P on the σ–algebra B∞ of Borel subsets of the product space, let us say Ω = {−1, +1}∞ , such that all the common laws of the projections are the same as those of Xn , n ∈ Z. The transformation T : Ω −→ Ω defined by T (ω)n := ωn+1 ; ω ∈ {−1, +1}∞ , is an invertible transformation preserving measure P. This defines a dynamical system. Conversely, given any dynamical system (Ω, F, P, T ) for every measurable function f : Ω −→ R1 we can define a stationary sequence of random variables Xn , n ∈ Z, by the formula Xn (ω) := f (T n ω); ω ∈ Ω. The transformation

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T induces an unitary operator UT : L2 (Ω, F, P) −→ L2 (Ω, F, P) defined by UT (f )(ω) := f (T ω); ω ∈ Ω, f ∈ L2 (Ω, F, P). Below we shall consider some properties of T like ergodicity, mixing, etc. or some spectral properties of the operator UT . One can find these notions in any book on basic ergodic theory. 3.

Finitary Sequences

We denote by M the set of all finite words of two letters H and T , by In the subset of words of length n ≥ 0 and by ∅ the empty word. The concatenation of two words is noted multiplicatively. It follows at once from the theorem of Kolmogorov that a function P : M −→ R induces a sequence of binary random variables Xn , n ∈ Z, if and only if P (x) ≥ 0, x ∈ M,

. . . (3.1)

P (∅) = 1,

. . . (3.2)

P (x) = P (xH) + P (xT ), x ∈ M.

. . . (3.3)

This sequence is stationary if and only if P (Hx) + P (T x) = P (x), x ∈ M.

. . . (3.4)

One can construct such a function P by induction. For arbitrarily fixed 0 ≤ p ≤ 1 we define P (H) := p, P (T ) := 1 − p. . . . (3.5) Let us suppose that we have already defined the function P for all words of fixed length n ≥ 1, satisfying the properties (3.1) − (3.4). For x ∈ In−1 we put: P (HxH) := αx , where 0 ≤ αx ≤ 1 is an arbitrarily fixed parameter. Following (3.3) and (3.4) we define: P (T xH) := P (xH) − αx , P (HxT ) := P (Hx) − αx , P (T xT ) := P (x) − P (Hx) − P (xH) + αx . This gives an extension of the function P to In+1 with the properties (3.2)−(3.4). To have (3.1) the parameter αx must satisfy max{0, P (xH) − P (T x)} ≤ αx ≤ min{P (xH), P (Hx)}, x ∈ In−1 . . . . (3.6)

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This interval is not empty ∈ In−1 , so there always exists an extension Sn for eachSxn+1 of the function P from k=0 Ik to k=0 Ik which satisfies (3.1) − (3.4) and therefore to M (Robertson, 1985). The corresponding stationary sequence of binary random variables Xn , n ∈ Z, is pairwise independent if and only if X P (HxH) = p2 , n ≥ 0. . . . (3.7) x∈In

Let η, ξ ∈ Rd be two fixed vectors such that η T ξ = 1. Let W be a d × d square matrix such that Wξ = ξ; η T W = η T . Let H be a d × d square matrix. Let us define:  if x = ∅,  I, M(x) := H, if x = H,  T := W − H, if x = T . Next, we define a function P : M −→ R by P (x) := η T M(x1 )M(x2 ) · · · M(xn )ξ, x := (x1 , x2 , . . . , xn ) ∈ In .

. . . (3.8)

Evidently this function satisfies (3.2) − (2.4). If a function P satisfies (2.1), then it determines a stationary sequence Xn , n ∈ Z, of binary random variables, which shall be called finitary of rank d (Robertson, 1985). The condition (3.1) is automatically satisfied if W ≥ H ≥ 0 (term by term) and η ≥ 0, ξ ≥ 0. If η T HHξ = p2 , . . . (3.9) where p is defined by (3.5), then for each n ∈ Z the random variables Xn and Xn+1 are independent. If WHξ = pξ, η T HW = pη T ,

. . . (3.10)

then for each n ∈ Z and k ∈ N the random vector (Xn , Xn+1 , . . . , Xn+k−1 ) and the random variable Xn+k+2 are independent; the same is true for Xn and (Xn+2 , Xn+3 , . . . , Xn+k+1 ). Property (3.10) (which is weaker than 1–dependence) and property (3.9) imply together the pairwise independence of the sequence Xn , n ∈ Z, associated to the function P . The common law of three successive random variables of this sequence is determined by the parameter αH := P (HHH), which must satisfy (3.6), i.e. max{0, (2p − 1)p} ≤ αH ≤ min{p2 , 3p2 − 3p + 1}.

. . . (3.11)

Example 1 (Robertson, 1985; Derriennic, 1991). For every fixed 0 ≤ p ≤ we are going to construct a particular sequence which satisfies a property stronger than (3.11): 3 1 3 p ≤ αH ≤ p3 . . . . (3.12) 2 2 1 2

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Let us define

1 1 1 1 , , , ), ξ T := (1, 1, 1, 1); 4 4 4 4 µ ¶ µ ¶ 0 B 0 C W := , H := ; A 0 D 0

η T := (

where: µ A :=

1 0 0 1



¶

, B := 

1/2 1/2

1/2



µ

 , C :=

1/2

u v

v u

u + v = p, 0 ≤ u ≤ p, 0 ≤ p ≤

¶

µ , D :=

2p 0 0 0

¶ ;

1 · 2

The matrix W is doubly stochastic and 0 ≤ H ≤ W. Moreover η T Hξ = p, η T Tξ = 1 − p. Next we have (3.10): µ WHξ =

BD 0 0 AC

¶



p p ξ= 0 0

0 0 0 0

0 0 u v

 0 0  ξ = pξ. v u

In the same way we obtain (3.7). Next, µ ¶ µ ¶ µ ¶ CD 0 2pu 0 2pu 2pv HH = , CD = , DC = , 0 DC 2pv 0 0 0 and we have (3.5). Finally, µ ¶ µ 0 CDC 2pu2 HHH = , CDC = DCD 0 2puv

2puv 2pv 2

¶

µ , DCD =

4p2 u 0 0 0

¶ ,

so η T HHHξ = 1/4(2p3 + 4p2 u) = 1/2p3 + p2 u. The parameter u varies in the interval [0, p], thus in this way we have constructed the required measure for αH satisfying (3.12). For p = 12 we obtain the sequence of Robertson and Womack 1 3 (1985) with 16 ≤ αH ≤ 16 · Robertson and Womack (1985) have proved for p = 12 3 and αH = 16 , that if we assume the property (3.10), then the corresponding sequence is unique. 4.

Constructive Case

Example 1 was defined by intermediary of the set of its marginal (finite dimensional) probability distributions. The next example is of a different nature – in its definition we use some given variables. One can imagine (and realize)

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easily such experience in practice, which is not so obvious for the preceeding case. It would be very interesting to verify if the eye could detect the difference between pairwise independent sequence and mutually independent sequence, if for the first one there is no Markov property of some degree (see Rosenblatt et al. 1962). Nevertheless this example is also finitary, as it shall be shown later. Example 2 (Bretagnolle et al. 1995). Let εn , n ∈ Z, be a sequence of mutually independent Rademacher random variables and let IAn , n ∈ Z, be a sequence of mutually independent and identically distributed indicator functions with P(An ) := α, 0 < α < 1. Moreover, we suppose that these two families {εn ; n ∈ Z} and {IAn ; n ∈ Z} are independent. We define new Rademacher random variables by: Xn := IAn εn + IAcn εn−1 εn−2 , n ∈ Z.

. . . (4.1)

This sequence is stationary and pairwise independent, but dependent by triplets (Bretagnolle et al. 1995). Evidently it is also 2–dependent. Theorem 1. There is no m ≥ 1 such that the sequence Xn , n ∈ Z, defined by (4.1) is a Markov chain of degree m. Proof. A direct calculation shows that for each n ∈ Z: P[Xn = Xn+1 = Xn+2 = +1] =

1 [1 + α2 (1 − α)], 8

1 [1 + α2 (1 − α)], 2 1 = +1] = [1 + 2α2 (1 − α)], 16

P[Xn = +1 | Xn−1 = +1, Xn−2 = +1] = P[Xn = Xn+1 = Xn+2 = Xn+3

1 α2 (1 − α) [1 + ], 2 1 + α2 (1 − α) ¤ 1 £ 1 + 3α2 (1 − α) , P[Xn = Xn+1 = Xn+2 = Xn+3 = Xn+4 = +1] = 32 P[Xn = +1 | Xn−1 = +1, Xn−2 = +1, Xn−3 = +1, Xn−4 = +1] · ¸ 1 α2 (1 − α) = 1+ , 2 1 + 2α2 (1 − α)

P[Xn = +1 | Xn−1 = +1, Xn−2 = +1, Xn−3 = +1] =

so the sequence Xn , n ∈ Z, is not a Markov chain of degree m = 3. For each fixed n ≥ 2 we have P[X1 = X2 = . . . = Xn = +1] =

1 2n E(

n Y

(1 + Xi ))

i=1

= 12 P[X1 = X2 = . . . = Xn−1 = +1] +

1 2n E(Xn

n−1 Y i=1

(1 + Xi )),

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so

½ E(X1 X2 X3 . . . Xk−1 Xk ) =

α2l (1 − α)l , 0,

for k = 3l; otherwise;

implies that for m ≥ 3 one has P[Xn = +1 | Xn−1 = +1, Xn−2 = +1, . . . , Xn−m = +1]   n−1 Q     E(X (1+X )) n i   i=n−m 1 = 2 1+ n−1 Q    E( (1+Xi ))    i=n−m

=

1 2

      

[m] 3

P

1+

α2i (1−α)i

   

i=1 [m] 3

1+

P

(m+1−3i)α2i (1−α)i

  

·

i=1

This expression depends on m, so we obtain the result. The sequence Xn , n ∈ Z, is a function of the following Markov chain Vn := (IAn , εn , εn−1 , εn−2 ) , n ∈ Z, of 16 states: (1,1,1,1) (1,1,1,-1) (1,1,-1,1) (1,1,-1,-1) (0,1,1,1) (0,1,-1,-1) (0,-1,1,1) (0,-1,-1,-1) (1,-1,1,1) (1,-1,1,-1) (1,-1,-1,1) (1,-1,-1,-1) (0,1,1,-1) (0,1,-1,1) (0,-1,1,-1) (0,-1,-1,1). The first eight states correspond to Xn = +1 and the last eight states correspond to Xn = −1. Let us define: a := α/2, b := (1 − α)/2. The transition matrix of this chain is given by:   a 0 0 0 b 0 b 0 a 0 0 0 0 0 0 0 a 0 0 0 b 0 b 0 a 0 0 0 0 0 0 0   0 a 0 0 0 0 0 0 0 a 0 0 b 0 b 0   0 a 0 0 0 0 0 0 0 a 0 0 b 0 b 0   a 0 0 0 b 0 b 0 a 0 0 0 0 0 0 0   0 a 0 0 0 0 0 0 0 a 0 0 b 0 b 0   0 0 a 0 0 0 0 0 0 0 a 0 0 b 0 b   0 0 0 a 0 b 0 b 0 0 0 a 0 0 0 0 W :=   0 0 a 0 0 0 0 0 0 0 a 0 0 b 0 b   0 0 a 0 0 0 0 0 0 0 a 0 0 b 0 b   0 0 0 a 0 b 0 b 0 0 0 a 0 0 0 0   0 0 0 a 0 b 0 b 0 0 0 a 0 0 0 0   a 0 0 0 b 0 b 0 a 0 0 0 0 0 0 0   0 a 0 0 0 0 0 0 0 a 0 0 b 0 b 0   0 0 a 0 0 0 0 0 0 0 a 0 0 b 0 b 0 0 0 a 0 b 0 b 0 0 0 a 0 0 0 0

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We denote by H the 16 × 16–matrix such that the first eight rows are the same as in W and the last eight rows are all composed of zeros. We define the matrix T by T := W − H. If ξ is the 16-dimensional vector having all coordinates equal to 1 and if η T is the right invariant probability vector of W: η T :=

1 (a, a, a, a, b, b, b, b, a, a, a, a, b, b, b, b), 4

then we have (3.1) − (3.3) and the probability distribution induced by (3.8) is equal to that of the sequence Xn , n ∈ Z, (see Robertson, 1973). We shall find a linear space generated by all the vectors of the form M(x1 )M(x2 ) · · · M(xn )ξ, where x1 x2 . . . xn ∈ In , n ∈ N. Let us denote: u1 u2 u3 u4 u5 u6

:= (1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0)T ; := (1, 1, −1, −1, 1, −1, −1, 1, 0, 0, 0, 0, 0, 0, 0, 0)T ; := (1, 1, 1, 1, 1, 1, −1, −1, 0, 0, 0, 0, 0, 0, 0, 0)T ; := (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1)T ; := (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, −1, −1, −1, 1, 1, −1)T ; := (0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, −1, −1, 1, 1)T .

The following matrix calculations are routine: Hξ = u1 ; Hu1 =

1 1 1 u1 + bu2 ; Hu2 = ( − b)u3 ; Hu3 = ( − b)u1 ; 2 2 2

1 1 1 u4 − bu5 ; Tu2 = −( − b)u6 ; Tu3 = ( − b)u5 ; 2 2 2 1 1 1 Hu4 = u1 − bu2 ; Hu5 = ( − b)u3 ; Hu6 = ( − b)u1 ; 2 2 2 1 1 1 Tu4 = u4 + bu5 ; Tu5 = −( − b)u6 ; Tu6 = ( − b)u5 . 2 2 2 Hence the linear space generated by {u1 , . . . , u6 } is invariant with respect to H and T, and therefore we can reduce our finitary system to the six–dimensional one given by:   1 0 α 1 0 α 0 0 −(1 − α) 0 0  1−α  1 0 α 0 0 α 0  A :=   1 0 α 1 0 α 2   −(1 − α) 0 0 1−α 0 0 0 −α 0 0 −α 0 Tξ = u4 ; Tu1 =

A+ has the first three rows of A followed by three rows of zeros, and A− := A − A+ . The left and right invariant vectors of A are ξ and η T , where η T :=

1 T (1, 0, α, 1, 0, α), ξ = (1, 0, 0, 1, 0, 0). 2

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Let us consider all vectors of the form η T As1 · · · Asn ; (s1 , . . . , sn ) ∈ {+, −}n , n ∈ N. We define η 1 := η;

η T2 :=

1 (0, 1, 0, 0, 1, 0); 2

η T3 :=

1 (1, 0, 0, −1, 0, 0). 2

Then we have η T1 A+ η T2 A+ η T3 A+ η T1 A− η T2 A− η T3 A−

= 41 (1, α2 , α, 1, α2 , α) = 21 η T1 + 21 α2 η T2 ; = 41 (1 − α, 0, 0, −(1 − α), 0, 0) = 21 (1 − α)η T3 ; = 41 (1, 0, α, 1, 0, α) = 12 η T1 ; = 41 (1, −α2 , α, 1, −α2 , α) = 21 η T1 − 12 α2 η T2 ; = 41 (−(1 − α), 0, 0, 1 − α, 0, 0) = − 12 (1 − α)η T3 ; = − 14 (1, 0, α, 1, 0, α) = − 12 η T1 ;

so we can restrict the transformations A+ and A− to the linear subspace generated by η 1 , η 2 , η 3 . Applying the results of Robertson (1988a) we obtain the following: Theorem 2. The sequence Xn , n ∈ Z, defined by (4.1) is finitary of rank 3. Its probability distribution is given by η˜T := ξ˜T := (1, 0, 0) and     1 0 1 1 0 −1 1 1 e :=  −α2 e :=  α2 H 0 0 , T 0 0 · 2 2 0 1−α 0 0 −(1 − α) 0 e are negative. From the It is also worth to observe that some entries of T above proof one can obtain: Corollary 1. If Xn , n ∈ Z, is a Rademacher sequence such that Xn = f (Yn ), n ∈ Z, for some recurrent finite–state Markov chain Yn , n ∈ Z, and for some ±1–valued function f , then it is finitary. e +T e = ξ˜η˜T proves that: The equality H Corollary 2. The sequence Xn , n ∈ Z, defined above is 1–dependent. One can ask some related questions: 1◦ The sequence Xn , n ∈ Z, was defined as a function of three successive members of some mutually independent and identically distributed sequence (which is called 3–block–factor). Can it be written as a function of two successive members of such a process (2–block–factor)? One knows that every 2–block–factor is one–dependent, but not conversely and all such examples are non–trivial. Recently Mat´ uˇs (1996) has proved the following result:

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A sequence Xn , n ∈ Z, is a 2–block–factor if and only if there exists a weakly exchangeable and completely dissociated double sequence Zij , i, j ∈ Z, (see Mat´ uˇs (1996) for the definitions) such that the common law of the overdiagonal Zi,i+1 , i ∈ Z, is equal to that one of the sequence Xn , n ∈ Z. If the answer for 1◦ is “yes”, it would be interesting to find such Zij , i, j ∈ Z, for the above sequence Xn , n ∈ Z. 2◦ What is the maximum of E(X1 X2 X3 ) for the class of 1–dependent and pairwise independent Rademacher sequences? For the last example we have E(X1 X2 X3 ) = 4/27 , if α = 2/3. One can deduce from (Robertson, 1985) that the maximum is strictly smaller than 1/2. 3◦ Is this sequence reversible i.e. for each n ∈ N the random vector (X1 , X2 , . . . , Xn ) has the same probability distribution as the random vector (Xn , Xn−1 , . . . , X1 )? 5.

New Examples

Now we shall give some new examples of sequences of pairwise independent Rademacher random variables. It is a routine job to verify those properties. Example 3. Let us take the sequences εn , n ∈ Z, and IAn , n ∈ Z, from Example 2. We define for fixed m ≥ 1: Xn(m) := IAn εn εn+1 . . . εn+m−1 + IAcn εn−1 εn−2 . . . εn−2m , n ∈ Z, and similarly for fixed m ≤ −1: Xn(m) := IAn εn εn+1 . . . εn+|m|−1 + IAcn εn εn−1 . . . εn−2|m|+1 , n ∈ Z. Example 4. Let M : Ω −→ Z be a random variable which is independent (m) of all variables Xn , n ∈ Z, m ∈ Z, defined above and such that P [M = 0] = 0, P [M = m] = P [M = −m] > 0, m 6= 0. (m) (0) ¯ n := P∞ We define Xn := 1, n ∈ Z, and X m=−∞ I[M =m] Xn , n ∈ Z. Example 5. Let us consider two sequences of centered signs εn , n ∈ Z; ηn , n ∈ Z; and two sequences of indicator functions: IAn , n ∈ Z; IBn , n ∈ Z; such that α := EIAn , n ∈ Z; β := EIBn , n ∈ Z.

We suppose that all those variables form a mutually independent family. Let us define for m ≥ 1 (m)

Xn

:= IAn ∩Bn εn εn+1 . . . εn+m−1 + IAcn ∩Bn εn−1 εn−2 . . . εn−2m +IAn ∩Bnc ηn ηn+1 . . . ηn+m−1 + IAcn ∩Bnc ηn−1 ηn−2 . . . ηn−2m , n ∈ Z;

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and for m ≤ −1 (m)

Xn

:= IAn ∩Bn εn εn+1 . . . εn+|m|−1 + IAcn ∩Bn εn εn−1 . . . εn−2|m|+1 +IAn ∩Bnc ηn ηn+1 . . . ηn+|m|−1 + IAcn ∩Bnc ηn ηn−1 . . . ηn−2|m|+1 , n ∈ Z.

¯ n , n ∈ N, as in Example 4. Next we define X One can generalise this example by taking a sequence of mutually independent and identically distributed partitions {An , Bn , Cn , Dn }, n ∈ Z, instead of two sequences of indicator functions. Example 6. The two preceeding examples are not ergodic. One can remove this default by modifying Example 4. First we lead the construction in such (m) a way that the families {Xn , n ∈ Z}; m ∈ Z, are mutually independent. Next we take a sequence of mutually independent and identically distributed random variables Mn : Ω −→ Z; n ∈ Z, which is independent of all variables (m) Xn ; n, m ∈ Z. Moreover, we assume that P [Mn = m] = P [Mn = −m] > 0, m 6= 0, P [Mn = 0] = 0. (m) ˜ n := P∞ We define X m=−∞ I[Mn =m] Xn , n ∈ Z. One can ask two questions: 1◦ Are the new examples finitary? 2◦ Do pairwise independent sequences of Rademacher (or binary) random variables which are not finitary exist? In particular, is Condition (3.12) necessary to be finitary?

6.

Simple Lebesgue Spectrum

The still unsolved problem of the existence of a dynamical system with simple Lebesgue spectrum is equivalent to the following probabilistic question: Does a stationary sequence of random variables Xn , n ∈ Z, defined on the common probability space (Ω, F, P) such that {1} ∪ {Xn ; n ∈ Z} form an orthonormal base of the Hilbert space L2 (Ω, F, P) exist? Even if such a system exists, it is not obvious that there also exists the one generated by a Rademacher sequence. But if it is the case, we have: Proposition 1. (Robertson, 1988; Womack, 1984) Let Xn , n ∈ Z, be a pairwise independent sequence of Rademacher random variables defined on some probability space (Ω, F, P). Then {1} ∪ {Xn ; n ∈ Z} is an orthonormal base of L2 (Ω, F, P) if and only if the variables Xn , n ∈ Z, generate the σ-field F and for each k > 1 we have the following equality +∞ X

[E(X1 Xk Xl )]2 = 1.

l=−∞

. . . (6.1)

new examples of pairwise independent random variables

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Condition (6.1) i.e. the identity of Parseval for the variables X1 Xk , k > 1, gives a criterion for “good” Rademacher sequences; for example, a sequence of mutually independent random variables is always “bad”. Fran¸cois Parreau has proved that orthonormal Walsh’s base (R´ev´esz et al., 1965) cannot be ordered in such a way that it becomes a stationary sequence. By Proposition 1, to obtain a “good” sequence it is sufficient to control the values of E(X1 Xk Xl ), −∞ < l < +∞, and to ensure the independence of sums in (6.1) for the parameter k > 1. Unfortunately, the general construction of pairwise independent random variables is not known till now and we have only a few particular examples of such sequences. Now we shall examine with respect to (6.1) all known examples of pairwise independent Rademacher random variables. For all previously known examples the sums in (6.1) were almost all zero. Now Examples 5 and 6 are such that all the sums considered are non–zero for all k > 1, but they are always smaller than 1 and theirs values depend on the parameter k. Example A. Mathew and Nadkarni (1984) have constructed an ergodic dynamical system which has the Lebesgue component in its spectrum with multiplicity two. It gives implicitly the first nontrivial example of a pairwise independent Rademacher sequence. It has the property that Xn , n ∈ Z, and −Xn , n ∈ Z, have the same distribution, so all its marginal laws are symmetric and the sums of (6.1) are all zero. Example 1. It is easy to verify that P[X1 = Xk = Xl = +1] =

1 {E(X1 Xk Xl ) + 1}, k, l ∈ Z, k 6= l 6= 1, 8

so in this case +∞ X

½ 2

[E(X1 Xk Xl )] =

l=−∞

(8αH − 1)2 ≤ 1/4 , 0,

if k = 2, 3, if k = 6 2, 3.

The corresponding dynamical system is ergodic, but not weakly mixing (Robertson, 1985; Robertson and Womack, 1988), therefore the maximal spectral type of UT is not continuous. Moreover, this sequence is reversible and it does not satisfy the zero–one law. The proofs of all these facts are the same as in (Robertson, 1985) for the case p = 1/2. Example B. Robertson (1988a) has constructed a pairwise independent Rademacher sequence Xn , n ∈ Z, with the property weaker than 2–dependence: for all n ≥ 1 the random vector (X1 , X2 , . . . , Xn ) and the random variable Xn+3 are independent.

. . . (6.2)

84

andrzej klopotowski and james b. robertson This sequence is finitary of rank 5. For each n ∈ Z we have  if k = n + 1, l = n + 2,  1/2 , 3 −1/4 , if k = n + 2, l = n + 4, αH = , E(Xn Xk Xl ) =  16 0, otherwise.

Then we obtain

  1/4 , 5/16 , [E(X1 Xk Xl )]2 =  0, l=−∞ +∞ X

if k = 2, if k = 3, if k 6= 2, 3.

This sequence is 4–dependent, so the corresponding dynamical system (being a K-system) has countable Lebesgue spectrum. It is not known if it is a function of Markov chain (in the sense of Corollary 1) or (at least) 5–block–factor. Example C. Robertson has given in (1988b) another sequence, which is 3 finitary of rank 14, satisfying all the assumptions of Proposition 1, αH = 16 and for each n ∈ Z:  if k = n + 1, l = n + 2,   1/2 ,    −1/4 , if k = n + 1, l = n + 5 or k = n + 4, l = n + 5, −1/2 , if k = n + 2, l = n + 4, E(Xn Xk Xl ) =   1/4 , if k = n + 2, l = n + 7 or k = n + 5, l = n + 7,    0, otherwise. Then we have

 5/8 ,     +∞ 7/8 ,  X 3/8 , [E(X1 Xk Xl )]2 =   1/8 , l=−∞    0,

if k = 2, if k = 3, if k = 5, if k = 6 or k = 8, otherwise.

This sequence is induced by some irreducible (recurrent) and periodic Markov chain (of period 4), so it is ergodic, but not weakly mixing (Robertson, 1988b). It is a function of the transitions of this chain (but not of its states), therefore it is a function of its two consecutive terms. It means that the partial converse to Corollary 1 is not true; one can have a finitary sequence which is not a function of the Markov chain defined by the matrix H + T. Nevertheless, this sequence is a function of another Markov chain with 16 states (Robertson, 1988b). Let us remark that this sequence is reversible and it has also a property weaker than 5–dependence: for all n ≥ 1 the random vector (X1 , X2 , . . . , Xn ) . . . (6.3) and the random variable Xn+6 are independent. The spectrum of the corresponding operator UT has a discrete component of the fourth roots of unity with simple multiplicity and a Lebesgue component with countable multiplicity.

new examples of pairwise independent random variables

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Example 3. For each fixed m ≥ 1 we have

(m)

= + + :=

(m)

(m)

X1 Xk Xl IA1 ε1 ε2 . . . εm · IAk εk εk+1 . . . εk+m−1 · IAcl εl−1 εl−2 . . . εl−2m IA1 ε1 ε2 . . . εm · IAl εl εl+1 . . . εl+m−1 · IAck εk−1 εk−2 . . . εk−2m IAc1 ε0 ε−1 . . . ε1−2m · IAk εk εk+1 . . . εk+m−1 · IAl εl εl+1 . . . εl+m−1 + · · · W1 + W2 + W3 + . . . + W8 .

One can see that EW1 = α2 (1−α) if k = m+1 and l = 2m+1; EW2 = α2 (1−α), if l = m + 1 and k = 2m + 1; and those terms are zero for any other values of k and l, EW3 = 0 for all k > 1 and finally EW4 = . . . = EW8 = 0 for all k, l ∈ Z. (m) (m) (m) Then for fixed m ≥ 1 we have E(X1 Xk Xl ) = α2 (1 − α), if k = m + 1, l = 2m + 1 or k = 2m + 1, l = m + 1 and this term is zero for any other values of k and l; therefore ½

+∞ X

(m) (m) (m) [E(X1 Xk Xl )]2 l=−∞

=

α4 (1 − α)2 , 0,

if k = m + 1, k = 2m + 1, otherwise. . . . (6.4)

This gives also the values of the considered sums for Example 2 (m = 1). In a (m) (m) (m) similar way for each fixed m ≤ −1 we have E(X1 Xk Xl ) = α2 (1 − α), if k = |m| + 1, l = 2|m| or k = 2|m|, l = |m| + 1 and zero in other cases. Hence +∞ X

½ (m)

[E(X1

(m)

Xk

(m)

Xl

)]2 =

l=−∞

α4 (1 − α)2 , 0,

if k = |m| + 1, k = 2|m|, otherwise,

i.e. this expression depend of k (and of |m|) as in (6.4). (m)

Let us remark that for each fixed m ∈ Z the sequence Xn , n ∈ Z, is |m| + 1–dependent, so the corresponding dynamical system has the countable Lebesgue spectrum. Example 4. The goal of this sequence is to remove the dependence of considered sums from the parameter m. We have ¯nX ¯k X ¯l = X

∞ X m=−∞

then for k > 1 one has

(m)

I[M =m] Xn(m) Xk

(m)

Xl

, n, k, l ∈ Z,

86

andrzej klopotowski and james b. robertson

¯l ¯k X ¯1X EX

=

=

P∞ m=−∞

(m)

P [M = m]E(X1

(m)

Xk

 = α2 (1 − α)P [M = k − 1],      α2 (1 − α)P [M = k−1 2 ], α2 (1 − α)P [M = −(k − 1)],   α2 (1 − α)P [M = − k2 ],    0,

(m)

Xl

)

if l = 2k − 1, if k odd and l = k+1 2 , if l = 2k − 2, if k even and l = k+2 2 , otherwise,

and finally we obtain ∞ X ¯1X ¯k X ¯ l )]2 [E(X l=−∞

½

2 if k odd, α4 (1 − α)2 {2(P [M = k − 1])2 + (P [M = k−1 2 ]) }, k 2 4 2 2 α (1 − α) {2(P [M = k − 1]) + (P [M £= ¤2 ]) }, if k even, = α4 (1 − α)2 {2(P [M = k − 1])2 + (P [M = k2 ])2 } < 1.

=

The assumption that this sum does not depend on k implies P [M = k] = P [M = 1] for each k > 1, which is impossible. Let us observe also that a ¯ n , n ∈ Z, is not ergodic; a partition into invariant sets is determined sequence X by the variable M . Example 5. Similar computations give ∞ P ¯1X ¯k X ¯ l )]2 = α4 (1−α)2 [β 3 +(1−β)3 ]2 {2(P [M = k −1])2 +(P [M = [E(X l=−∞ £k¤ 2 2 ]) }. One can see that these sums are smaller than those of Example 4. Example 6. We have verified that for k > 1 one has · ¸ ∞ X k 6 2 4 2 6 ˜ ˜ ˜ [E(X1 Xk Xl )] = α (1 − α) {2(P [M1 = k − 1]) + (P [M1 = ]) }, 2 l=−∞

which gives also the sums smaller than those of Example 4. Since this sequence is a function of an infinite product of Bernoulli shifts, then it is also a Bernoulli shift. Consequently it is mixing of all orders. Remark 1. Using some minor modifications of these examples for every 4 fixed k and l and for every 0 < |ρ| ≤ 27 one can find a pairwise independent 4 Rademacher sequence Xn , n ∈ Z, such that E(X1 Xk Xl ) = ρ. For |ρ| > 27 and 1 especially for |ρ| = 2 the situation is not clear at all. Even very simple questions like: Is it possible to have E(X1 Xk Xl ) = 12 for two different pairs of k and l? Is it possible to have E|(X1 Xk Xl )| = 12 for three different pairs of k and l? ... and so on, seem to be very difficult to answer.

new examples of pairwise independent random variables

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Remark 2. From the point of view of the ergodic theory it is easy to understand, why our research was not successful. All these sequences (except Example A) have positive entropy and those with the simple Lebesgue spectrum must have the entropy zero. The sums in (6.1) are non zero because of some asymmetry of their probability distributions. Example A is perfectly symmetric. It would be very interesting to find some asymmetric pairwise independent Rademacher sequences with the entropy zero. Remark 3. Let us suppose that one can find a sequence of Rademacher random variables εn , n ∈ Z, four by four (but not five by five) independent and such that the entropy of the corresponding dynamical system is zero. Let IAn , n ∈ Z, be stationary sequence of indicator functions with P(An Acn−1 Acn−2 ) 6= 0, n ∈ Z. If those two families {εn ; n ∈ Z} and {IAn ; n ∈ Z} are independent, then the sequence Xn , n ∈ Z, defined by (3.1) is Rademacher and is pairwise independent, but not three by three. This sequence is not necessarily 2–dependent. Is it possible to find “natural” properties of IAn , n ∈ Z, so that the entropy of the dynamical system corresponding to Xn , n ∈ Z, is zero? Acknowledgements. The revised version of this paper was prepared while the first author was a Visiting Fellow at Centre of Advanced Study in Mathematics, University of Mumbai. He expresses his deep gratitude to Prof. M. G. Nadkarni and all Staff of Departments of Mathematics and Statistics for a hearty reception. References

Bretagnolle, J. and Klopotowski, A. (1995). Sur l’existence des suites de variables al´ eatoires s a s ind´ ependantes ´ echangeables ou stationnaires, Ann. Inst. Henri Poincar´ e, 31, 325–350. ´ n, C. (1991). On the asymptotic behavior of sums of pairwise Cuesta, J.A. and Matra independent random variables, Statistics and Prob. Letters, 11, 201–210. Derriennic, Y. (1991). A letter. Derriennic, Y. and Klopotowski, A. (1988). Variables al´ eatoires deux a deux ind´ ependantes. I. Cas de trois variables binaires II. Cas de quatre variables binaires, Publications de L.A.N.S. INSA de Rennes. − − −− (1991a). Cinq variables al´ eatoires binaires stationnaires deux ` a deux ind´ ependantes, Pr´ epubl. Institut Galil´ ee, Universit´ e Paris XIII, 1–38 − − −− (1991b). Sur les hypoth` eses constructibles concernant des suites de variables al´ eatoires binaires, idem, 1–10 Mathew, J. and Nadkarni, M.G. (1984). A measure preserving transformation whose spectrum has Lebesgue component of multiplicity two, Bull. London Math. Soc., 16, 402–406. ´ˇ Matu s, F. (1996). On two-block-factor sequences and one-dependence, Proc. Amer. Math. Soc. 124, 1237–1242 ´ve ´sz, P. and Wschebor, M. (1965). On the statistical properties of the Walsh functions, Re Publ. Math. Inst. Hung. Acad. Sci., 9A, 543–554. Robertson, J.B. (1973). A spectral representation of the states of a measure preserving transformation, Z. Wahrsch., 27, 185–194. − − −− (1985). Independence and Fair Coin-Tossing, Math. Scientist, 10, 109–117.

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− − −− (1988a). A two state pairwise independent stationary process for which X1 X3 X5 is dependent, Sankhy¯ a, Series A, 50, 171–183. −−−− (1988b). Another pairwise independent stationary chain, Approximation, Probability and Related Fields. Edited by G. Anastassiou and S.T. Rachev, Plenum Press, New York 423–434. Robertson, J.B. and Womack, J.M. (1985). A pairwise independent stationary stochastic process, Statistics and Probability Letters, 3, 195–199. Rosenblatt, M. and Slepian, D. (1962). N -th order Markov chain with every N variables independent, J. SIAM, 10, 537–549. Womack, J.M. (1984). Pairwise independent stationary stochastic processes, Ph. D. Thesis, University of California, Santa Barbara 1–81.

Andrzej Klopotowski ´ Paris XIII Universite ´e Institut Galile ´ment av. J.-B. Cle 93430 Villetaneuse cedex France e-mail : [email protected]

James B. Robertson Department of Statistics and Applied Probability University of California Santa Barbara California 93106-3110 USA e-mail : [email protected]