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The central limit theorem for Tukey's 3R smoother
Statistics & Probability North-Holland
2 January
Letters 13 (1991) 29-37
The central limit theorem smoother Wlodzimierz Department
Bryc and Magda Peligrad
of Mathematical
Sciences,
University
1992
for Tukey’s 3R
*
of Cincinnati, Cincmnati, OH 45221-0025, USA
Received May 1990 Revised May 1991
Abstract: Asymptotic properties of a non-linear smoothing algorithm are analyzed. It is shown that under appropriate output of the algorithm is absolutely regular and that partial sums are asymptotically normal. AMS
1980 Subject Ckzssifications:
Keywords:
3R smoother,
Primary
60F05;
weak dependence,
Central
Secondary
conditions
the
60GlO.
Limit Theorem.
1. Introduction and notation In his discussion of exploratory analysis of time series Tukey (1977) defined and advocated a non-linear smoothing algorithm in which the running median of three consecutive quantities is applied repeatedly until no further change occurs. The algorithm, called the 3R smoother, has the properties of being resistant to occasional outliers and at the same time responsive to changes in trend. Mallows (1979, 1980) discussed some theoretical aspects of the 3R smoother for strictly stationary sequences of random variables and he also raised the question as to under what conditions the output of the algorithm is either strongly (Rosenblatt) mixing, or +mixing; see Mallows (1979, p. 83). Bradley (1984) answered both questions, showing that: (i) the ouput of the 3R smoother is strongly mixing, provided the input is; he gave also explicit bounds for the convergence rate of the strong mixing coefficients. (ii) the output of the 3R smoother may fail to be +mixing even for an i.i.d. input. In this paper we prove that the absolute regularity condition is preserved by the 3R smoother, complementing (i) above. We also have something to say about case (ii): we use the ‘two part mixing’ conditions, see Bradley and Peligrad (1986), to analyze the dependence structure of the output of the 3R smoother if the input is a +mixing strictly stationary random sequence. As a consequence we show that if the input sequence is +mixing then even though the output of the 3R smoother may fail to be $-mixing, under appropriate moment conditions we have enough information to establish the Central Limit Theorem. Furthermore, if the input sequence is i.i.d., then we show that the output of the 3R smoother satisfies the Central Limit Theorem with the rate of convergence close to the best possible, even though the input sequence is not assumed to have second moments. None of those Central Limit Theorems could be obtained from the previously known fact that the strong mixing condition is preserved by the 3R * Supported
in part by NSF grants
0167-7152/92/$05.00
DMS-8905615
and DMS-9007986.
0 1992 - Elsevier Science Publishers
B.V. All rights reserved
29
STATISTICS & PROBABILITY
Volume 13, Number 1
LETTERS
2 January 1992
smoother. Our Lemma 5.2 below is also of independent interest, see condition A5 in Mallows (1980, p. 701). To give the formal definition of the 3R smoother, consider the running median mapping R : RE ---)Iw‘, defined for x = (x~)~ E E by R(x) ByRN=Ro
= (med(-+,,
xk?
Xk+l))k~E~
0 R we denote the n th iteration X0 is defined as
. . .
with the input
X” = lim.,,
(1)
(composition)
of R. The output
Xm of the 3R smoother
R"(X')
(2)
provided the limit exists (on each coordinate). From discussion in Tukey (1977), Mallows (1979) and from Theorem B in Bradley (1984) it follows that the limit (2) exists a.s. and is stationary and mixing, provided the input X0 = (X,“) is strictly stationary and mixing in the ergodic theoretic sense. Notice that when applied to finite sequences of data, the algorithm is usally modified by discarding end-point observations in each iteration, see e.g. Tukey (1977, pp. 210-213). This does not lead to significant loss of information, as the algorithm usually reaches fixed point in several iterations; the last claim is supported both by the numerical evidence on several sets of real data in Tukey (1977) and by theoretical estimates (3.1) and (3.2) below. It might be also worth pointing out that the 3R smoother algorithm with an i.i.d. input is an example of the ‘cellular automaton’, see e.g. Schonmann (1990) for another ‘transition rule’ and references. Except in the simplest case d = 1, little seems to be know about asymptotic behavior of variants of the 3R smoother defined for general Zd-indexed (even O-l valued) i.i.d. input sequences. We shall be interested in the strong mixing properties of the output X”. Those will be expressed in terms of the strong mixing coefficients defined as follows: be a strictly stationary sequence of random variables. Define past and future u-fields Let x=(xk>kGz 9,(X) = u( X1 : j < k) and pn( X) = a( X, : j > n) (the dependence of u-fields on X will be omitted, if no misunderstanding results). Put CX,(X)=SU~{(P(A~B)-P(A)P(B)):AEP~(X), (P,(x)=sup{IP(BIA)-P(B)I:AE~o(x),
4??(X)= sup
P(A n B)
il
P(A)P(B)
- 1 : A Ego(X),
BEF~(X)}, BE%(X),
BEE(X),
(3)
%+O},
P(A)P(B)
(4)
> 0 ) 1
i,J where the supremum in (6) is taken over all finite PO(X)-measurable partitions {A;} and over all finite sn (X)-measurable partitions { B, }. A stationary sequence X = (Xk) is called: - strongly mixing, if a,(X) -0 as n -+ cc; - +-mixing, if +n(X) + 0 as n ---f co; - +mixing, if q,(X) + 0 as n --j co; - absolutely regular, if p,(X) + 0 as n + 00. It is known that 0 G a,(X) G /I,(X) G I&(X) G 4,(X). Dependence of left hand sides of (3)-(6) on X will be omitted, if no misunderstanding results. ‘ e as m + 00’ instead of O(m); w denotes the standard Brownian In this paper we use notation motion on [O,l], [x] denotes the integer part of x.
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2. Results Theorem 1. Suppose that X is a stationary absolutely regular randon sequence. Furthermore there is 0 < q < 1 and C > 0 such that for all n, k >, 1, P&k+4(Xa))
G P,,,(X)
+ CW
Then X” = Rm(X)
exists.
” %1(X)).
(7)
(Our proof gives C = 4(1 - q)-‘,) The following corollary lists some of the consequences of Theorem 1; in particular, it shows that the absolute regularity is preserved by the 3R smoother and explicit bounds for the rate of convergence to 0 are found. Corollary (i) Zf (ii) Zf (iii) If n -+ co. (iv) If
1. If X p,,(X) p,(X) p,,(X)
is + < <
a strictly stationary random sequence then the following implications hold: 0 as n -+ co, then X” = R”(X) is defined a.s. and &(Xm) + 0 as n ---) 00. ne8 as n + 00 for some 8 > 0, then /3,(X”) -=z nPej2 asn-+co. pn as n + co for some 0 < p < 1 then there is 0 < q < 1 such that /3,(X”) < qfi
X is m-dependent,
then j3,(Xcc)
Since it is known that aperiodic Harris (1986), we get also the following result.
as
+z p” for some 0 -=zp < 1. recurrent
Corollary 2. If X = (X,), rZ is an aperiodic X” = R”(X) exists and is absolutely regular.
Markov
Harris
chains
recurrent
are absolutely
stationary
Markov
regular,
see e.g. Bradley
chain,
then the output
q
From Corollary 1, using limit theorems known in the literature, one can deduce limit theorems for partial sums of X” under appropriate assumptions on a rate of convergence to 0 of b,(X) and under some additional moment assumptions. However, if the input sequence is $-mixing, we do not need to impose any conditions on the rate of convergence of Q,,(X) to 0. This is achieved by using the approach developed in Bradley and Peligrad (1986), together with the following result. Theorem 2. Suppose that X is a strictly stationary a.s. and there are C < co, 0 6 p < 1 such that IP(A for allA
f-l B) - P(AP(B)
ELP,,(X~),
IG
G-mixing
~[.,2-4](X)f$4)
random sequence.
+
CP”
is defined
(8)
BEAM.
The following result establishes the Central Limit Theorem sequence; no mixing rate assumption is required. Let ST=
Then X* = R*(X)
i:
xr,
under
G-mixing
dependence
of the input
u,’ = Var( Snm).
k=O
Theorem 3. Let X be a strictly stationary $-mixing random sequence. Suppose E{ 1X,, ) 2ts} -C co for some 6 > 0. Then X” = R”(X) exists, and E{ 1X,M ) ‘+‘} < co. If in addition for each E > 0, u,‘/n’-’
+ 00
(9) 31
Volume 13, Number 1
STATISTICS & PROBABILITY LE’M’ERS
as n + CO. then (Snw - nE{ Xr })/a,, + N(0, 1) in distribution
2 January 1992
as n + 00. Moreover,
(10) weakly in D[O,l] as n + co. Furthermore, if X is symmetric then E{ XF} = 0.
(i.e., ( XP)P E z and ( - XP)P E z have same finite dimensional
distributions),
The following theorem shows that if the input is an i.i.d. sequence, the moment assumptions for the Central Limit Theorem can be significantly relaxed. In particular the CLT holds even when the input sequence does not have finite second moments. Furthermore, the rate of convergence in the Central Limit Theorem can be analyzed, and (9) can be dropped. Theorem 4. Suppose X is an i.i.d. non-degenerate random sequence such that E{ 1X0 )log’ +&12(1 + 1X0 I)} < cc for some 6 > 0. Then X” = R”(X) exists and E{ ] X7 I2 log2+“(1 + ( X,M I)} < co. Furthermore o,‘/n --$ a2 > 0 as n + M and (S,m - nE { Xr })/an’12 -+ N(0, 1) in distribution as n + 00. Suppose in addition that E{ 1X,, ( 3/2 } < 00. Then there is a constant B = B(X) such that < Bn-7’2
-CX Moreover,
if X,-, is symmetric,
then XF IS symmetric
log’ n.
and hence E { XT}
Remark. If for some 0 the distribution of X,, - 8 is symmetric, symmetric; in particular Theorem 4 allows to write the normal has no second moments. Theorems 1, 2 and the corollaries and 5 respectively.
are proved
in Section
= 0.
then the same proof shows that XF - 13is confidence interval for 8 even when X0 - 8
3. Theorems
3 and 4 are proved
in Sections
4
3. Proofs of Theorems 1 and 2 Our proof monotonic
follows the general idea of Bradley (1984). An ordered triplet of numbers if either a 6 b < c or c < b < a holds. Define the ‘random times’: T+=inf{t>,O:(X,_,, T-=inf{s>O:(X_,_,,
X, , X, + , ) is monotonic} X_,,
X_,+,)
(a,
b, c) is called
,
ismonotonic)
We shall show that that for every n, m, k >, 1, P(T”
>, n + m + k) < P(T”
>, n)P(TO
> m) + CY~(X),
P(T”
>, n + m + k) < P(T”
>, n)P(T”
> m) + $,(X)P(T”
(11) >, n),
(12)
where To is either T+ or T-. To prove (11) and (12) we shall consider the case To = T+. Since T- is the same as Tt for the time-reversed sequence, the proof for the case To = TP is similar and is omitted.
magnitude. In particular Observe that Tf > n means that the numbers X0, X,, . . . , X, _ 1 are of alternating and T+ > n + m + k implies the occurrence of both events A = { T+ > n } and T+>, n is .%$_, -measureable 32
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Volume 13, Number 1
2 January 1992
LETTERS
X,, X,,,) for any n + k G r Q n + m + k}. Hence P(T” B = {there is no monotonic triple (X,-i, using (3) or (4) respectively we get + k) G P(A n B). Since A Ed,_, and B E.S$+~-,(X),
>, n + m
P(AnB)~P(A)P(B)+cu,(X), P(A
n B) G P(A)P(B)
+G,(X)P(A).
By stationarity P(B) = P(T o > m). Therefore (11) and (12) follow. We shall also need the following observation: If X is strictly stationary and mixing (in the ergodic theoretic sense), then P(( X_,, X,,, X,) is monotonic) > 0, see Bradley (1984, p. 254). In particular P(T” an) G P(T” 2 1) = 1 - P((X_,, X0, X,) is monotonic) -C 1 for every n 2 1. Put q = 1 - P(( X_,, The following
X0, X,) is monotonic).
two lemmas
(13)
make use of (11) and (12) to estimate
tails of random
variables
T+ and T-.
Lemma 3.1. For every n, k >, i, P(T”
> nk) < 2(1-
q)-‘(a,_,
V
(14)
q”),
where 0 < q < 1 is defined by (13) and cxk = CQ(X)
is defined by (3).
Proof. Since P(T o 2 k) < q, from (11) we get P(T”
>, (a + 1)k)
> nk) + CQ_,.
Therefore P(T”&nk)
...
<2(1-q)-1(q”vcu,_,).
+q(Cy,_,+q2)...)=qn+cx_l
‘;g;’
0
Lemma 3.2. If $” + 0, then there are 0 Q fi < 1 and d < cc such that P(TO>n)<@
foreueryn>l.
Proof. Let k 2 1 be such that 0 = c@~_,+ q < 1, where q is defined P(T o 2 (n + 1)k) G 0P(T o > nk). Hence P(T o > nk) 6 8” and the
C=eP.
by (13). Inequality (12) implies conclusion holds with F = 61’k,
0
We shall also use the following Lemma A. ZfrfA~a(R~(X),:O
B
lE+,v+i
of Bradley
(1984) Lemma
j< -n), thenAn{T-
The proof of both Theorems A EEn-,?-,
consequence
1 and 2 is now concluded
(X”)ca(Rk(X),:j<
as follows:
2.4.
0 Fix N, k, n > 1 and let
-n-N-l),
(X”)~cr(R~(X),:j>n+N+l). 33
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Define TN =inf{t>O:(X,+,_,,
XrfN,
T; =inf{t>O:(X_,_._,,
X,,,,,)
Xmr-N,
ismonotonic},
XPr-N+,
) is monotonic}.
Clearly TN+ and T; satisfy the conclusions of Lemma 3.1 and Lemma 3.2. Indeed, and T- respectively for shifted stationary sequences (Y,), t L = (X, * N)k E E. We have
they are equal to T+
{T;~n}nBn{T,i~n})
P(An
~P(AnB)~P(An{T~~n}f3BB{T~fn})+P(max{T~,TT,-}>n).
(15)
By Lemma A applied to the shifted sequences (Y,), E z = (X, * N)k E =, we have P_,+,(X) and B n {T,,, G rz}E.F&~(X).Therefore
n {T;GH} nsn
P(A
A
n {T; G n } E
{TN+
G'(A~{T,-Q~})P(B~{T,+~})+~,,~,(X)P(A~{T,-~~}) ~fyA)~(B)+(P,,-,~(A). Lemma
3.2. together
with (15), (16) gives therefore
nB)-P(A)P(B)<+,,_,P(A)+
P(A
(16)
2&n+‘.
(17)
Since for a complement B' of B we have P(A nB')- P(A)P(B')=P(A)P(B)- P(A nB), inequality (8) follows for n = 4k + 2 from (17) by picking N = n = k in (17). The general case then follows by the standard argument, yielding C = 2@j2 and p = $/4. 0 Similarly we prove Theorem 1. Let { Ai) be a R n _ v_ 1(X*)-measurable partition. Then F n+ N+, (X”)-measurable ;C/P(AinB,) i,j
partition
and let { B, > be an
-%4)+,)/
<:CIP(A,n
{T;
nB,n {T~~n})-P(A,n{T',
i.i ++xP(A,nB,n{T,f i,i
>n})+:~P(A,nB,n{T,->n}) i,i
+iCP(A,)f'(B,n(T,t >n})+;~P(A,n{T,-w})P(B,) i,i i,i
nk for n and putting P 4&+4(Xm)
GP,,,(X)
N = nk + 1 we get
+40 -~)-l(%-lVcf)~
q
which is a Proof of Corollary 1. All cases except (iv) follow from /34n~+4(X”) G (C + 1)q” V /3,-l(X), trivial consequence of (7) applied with k = n. To obtain (iv) we choose k > 1 such that cy,_,(X) = 0. Then for n 2 1, Dznk = 0 and by (7) we have P 4n/c+4 34
G
cc
+
lh”.
0
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1992
4. Proof of Theorem 3 The following
simple observation
med( -a,
-b,
-c)
is a consequence = -med(a,
Lemma 4.1. Zf R”( X) exists, then R”( -X) 0 is a symmetric random variable. The next observation
of the fact that
b, c). = -R”(X).
shows that the 3R smoother
Lemma 4.2. Zf X is a stationary
In particular,
preserves
4.1, inequality
P((X,“12t)63P((XO(&t)
(18)
(18) implies
XF > t implies
that at least one of the variables
also that P( Xr < t) < 3P( X0 < t) for every t E R;
foreveryt20.
(19)
Proof of Theorem 3. We shall apply Theorem 1 in Bradley and Peligrad (1986) By (8), condition (1.5) of Bradley and Peligrad (1986) is satisfied (X?)kEZ. geometrically. Indeed for every A E .FY,,(Xco), B E F2,+ s( Xm), n@
-P(A)P@)J
G /p(B)/\&4 < /P(B)
then XT
exists then
foreveryttE.
Proof. Indeed, it is enough to observe that the inequality X _,, X0, Xi cannot be smaller than t. 0
IP(A
in distribution
the moments.
sequence of random variables such that R”(X)
P(X,“>t)<3P(X,>t)
Remark. In view of Lemma in particular
if X G -X
n@
to stationary sequence with {a,} decreasing
-P(A)P@)\
+,,J’( A) + Cp2n G &&GQ~~
+ GP”.
By (19) we have
E{ 1x7 12+6} = (2 + fi)/atl+aP(
1X,m ) > t) dt < 3E( ) X,, )*+a} < CO.
0
Since by assumption 0,:/n' -’ + co as n ---*CO for every E > 0, (10) follows from the conclusion and Peligrad (1986). If in addition X0 is symmetric, then E{ XT} = 0 by Lemma 4.1. q
of Bradley
5. Proof of Theorem 4 Recall that the sequence (X,), E z is associated, see Esary, Proschan and Walkup (1967) if for each m 2 1 and every bounded continuous coordinatewise nondecreasing functions f, g : R*“‘+’ + Iw we have Cov(f(X-, ,..., X,), 8(X_, ,..., X,)) >, 0. Lemma 5.1. Zf X is i.i.d.,
then R”(X)
exists and is associated.
Proof. Indeed, X (being i.i.d.) is associated and since x, y, z + med(x, y, z) is the nondecreasing function of each coordinate, therefore R”(X) is associated for each n 2 1. The limit R”(X) exists because 35
Volume
13, Number
1
STATISTICS
of the i.i.d. assumption 0
& PROBABILITY
and hence is also associated,
2 January
1992
(1967) property
P5.
LETTERS
see Esary, Proschan
and Walkup
Remark. The proof of the lemma shows that if X is associated, then so is X”. In particular 2 0 for every k 2 1 and lim inf, _ ,n-’ Var(C, d n XT) > 0 in the nondegenerate case.
Cov( X7,
XF)
Lemma 5.2. Suppose f ( .) 2 0 is a nondecreasing and differentiable function, f(x) Z 0 for x f 0 and X is an i.i.d. sequence such that E { f ( l X0 1)) < co. Then the output X” = R”(X) satisfies E{ f2( I Xl
I>} < CQ.
(20)
Proof. Let (XL),,, = R(X). Since XA > t implies that at least two of the variables therefore by the i.i.d. assumption P(Xi 2 t) < 3P2(X0 > t). Lemma 4.1 gives
X_,,
X0, Xi exceed t,
P( I x; (2 t) < 3p2( I x, I 2 1) for each t 2 0. This, together
(21)
with (19) implies
P( 1x,m I > t) f 9P2( I x, ) > t). Since by Chebyshev’s known tail integration
inequality formula:
(22)
P( l X,, l > t) G E{ f( l X0 ])}/f(t),
(20) follows
from (22) and the well
sc+18E{f(lX,l)}~mf~(t)~(lXol~t)dt~~.
Remark. Inequality
(22) can be complemented
P( 1x; 1 > t) ,, Indeed
XT
:P’( 1x,
•I
with the lower bound
1 > t).
> t holds on { X0 > t } f~ { X, > t } and Lemma
(23) 4.1 ends the proof.
Remark. The constant in (22) can be improved to 9 by using Mallows (1979) formula for the distribution of X$‘. Lemma 5.2 holds also for +mixing input sequences X which satisfy q,(X) < cc (with the constant in (21) replaced by 3(1 + $i(X))). The condition #,(X) < cc is satisfied in particular for Markov chains on ([E, .G@)with one step transition probabilities P(x, A) such that there is C < cc satisfying P(x, A) G CP( y, A) Vx, y E E, VA E .?8 (such Markov chains were studied e.g. in large deviation theory). We shall also need the following Lemma 5.3. Suppose c%(Y)
Y = (Y,)
= P”
E{ q2 log’+‘(l
lemma.
is strictly stationary,
forsome
O
+ I r, I)} < co.
Assume also E{ Y,} = 0 and put S,, = Ck <“Yk, u,’ = Var(S,). Then n-’ Moreover, if CJf 0, then S[nt]/u, + w weakly in D[O,l] as n + 00. 36
(24) (25) a~*+u2forsome
a’>,0
asn+oo.
Volume
13, Number
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& PROBABILITY
LETTERS
2 January
Proof. First by Lemma 1 in Herrndorf (1985) specialized to the function G(x) = x*(log(x + l))‘+‘, S > 0, and with f,(x) = 2i+’ llog x 1-u+‘) as x + 0, x > 0, we can find a constant C such that IE{Y,~})I~Clloga,I-(‘+“)(E{Yo’ Therefore
log’+s~l+Y,l})2
the sum u2 = E{ Yi} + 2 CyZ1 E{ Y,?} Var
=nE{Y;}
+2
5
converges.
(n-j)E{Y,Y,}
1992
where
foreveryial.
Now by stationarity =no*(l
+0(l))
asn+cc,
j=l
which proves the first part of the lemma. The second part is now a particular after Theorem 2 in Herrndorf (1985). 0
case of the corollary,
stated
Proof of Theorem 4. In order to prove the first part of Theorem 4, we have only to verify the assumptions of Lemma 5.3. As follows from Theorem 1 (or Bradley, 1984, Corollary l(iv)), (XF), =z is stationary and strongly mixing with (Y, < p” for some 0 < p < 1, which proves (24). By Lemma 5.2 applied with f(x) = x log’ +&‘*(1 + x), condition (25) is satisfied. To verify that u f 0, notice that by Lemma 5.1 (XF) is an associated sequence and in particular lim inf, _ m Var(E.jl= i X:)/n > Var( XF). From the lower bound (23) we have Var( XT) # 0, provided X0 is not a constant. Therefore u Z 0 and the first part of Theorem 4 follows now from Lemma 5.3. The second part of Theorem 4 follows from Theorem 2.1 in Birkel (1988) applied to stationary sequence ( Xp)k E z. Indeed, (Xp) is associated by Lemma 5.1. Moreover, by Lemma 5.2 we have E{ 1XT 13} < cc; therefore assumption (2.3) in Birkel (1988) is satisfied. Since it is well known that Cov(X,“, X;) < [a,JX”)J”3(E{ I X,m 13}>*I3 by Theorem 1 (or Corollary l(iv) in Bradley, 1984), condition (2.1) in Birkel (1988) is satisfied. Therefore the conclusion of Theorem 2.1 in Birkel (1988) ends the proof of Theorem 4. q
References Birkel, T. (1988) On the convergence rate in the Central Limit Theorem for associated processes, Ann. Probab. 16, 16851698. Bradley, R.C. (1984), Some mixing properties of Tukey’s 3R smoother, Stochastics 11, 249-264. Bradley, R.C. (1986) Basic properties of strong mixing conditions, in: E. Eberlein and M.S. Taqqu, eds., Dependence in Probability and Statistics (Birkhauser, Boston, MA). Bradley, R.C. and M. Peligrad (1986) Invariance principle under a two part mixing assumption, Stochastic Process Appl. 22, 211-289. Esary, J.D., F. Proschan and D.W. Walkup (1967), Association of random variables with applications, Ann. Math. Statist. 38, 1466-1474.
Hermdorf, N. (1985), A functional central limit theorem for strongly mixing sequences of random variables, Z. Wahrsch. Verw. Gebiete 69, 541-550. Mallows, C.L. (1979), Some theoretical results on Tukey’s 3R smoother, in: T. Gasser and M. Rosenblatt, eds., Smoothing Techniques for Curve Estimation. Lecture Notes in Math. No. 757 (Springer, Berlin). Mallows, CL. (1980) Some theory of non-linear smoothers, Ann. Statist. 8, 695-715. Schonman, R. (1990), Critical points of two dimensional bootstrap percolation like cellular automata, J. Statist. Phys. 58, 1239-1244. Tukey, J.W. (1977) Exploratory Data Analysis (Addison-Wesley, Reading, MA).
31
2 January
Letters 13 (1991) 29-37
The central limit theorem smoother Wlodzimierz Department
Bryc and Magda Peligrad
of Mathematical
Sciences,
University
1992
for Tukey’s 3R
*
of Cincinnati, Cincmnati, OH 45221-0025, USA
Received May 1990 Revised May 1991
Abstract: Asymptotic properties of a non-linear smoothing algorithm are analyzed. It is shown that under appropriate output of the algorithm is absolutely regular and that partial sums are asymptotically normal. AMS
1980 Subject Ckzssifications:
Keywords:
3R smoother,
Primary
60F05;
weak dependence,
Central
Secondary
conditions
the
60GlO.
Limit Theorem.
1. Introduction and notation In his discussion of exploratory analysis of time series Tukey (1977) defined and advocated a non-linear smoothing algorithm in which the running median of three consecutive quantities is applied repeatedly until no further change occurs. The algorithm, called the 3R smoother, has the properties of being resistant to occasional outliers and at the same time responsive to changes in trend. Mallows (1979, 1980) discussed some theoretical aspects of the 3R smoother for strictly stationary sequences of random variables and he also raised the question as to under what conditions the output of the algorithm is either strongly (Rosenblatt) mixing, or +mixing; see Mallows (1979, p. 83). Bradley (1984) answered both questions, showing that: (i) the ouput of the 3R smoother is strongly mixing, provided the input is; he gave also explicit bounds for the convergence rate of the strong mixing coefficients. (ii) the output of the 3R smoother may fail to be +mixing even for an i.i.d. input. In this paper we prove that the absolute regularity condition is preserved by the 3R smoother, complementing (i) above. We also have something to say about case (ii): we use the ‘two part mixing’ conditions, see Bradley and Peligrad (1986), to analyze the dependence structure of the output of the 3R smoother if the input is a +mixing strictly stationary random sequence. As a consequence we show that if the input sequence is +mixing then even though the output of the 3R smoother may fail to be $-mixing, under appropriate moment conditions we have enough information to establish the Central Limit Theorem. Furthermore, if the input sequence is i.i.d., then we show that the output of the 3R smoother satisfies the Central Limit Theorem with the rate of convergence close to the best possible, even though the input sequence is not assumed to have second moments. None of those Central Limit Theorems could be obtained from the previously known fact that the strong mixing condition is preserved by the 3R * Supported
in part by NSF grants
0167-7152/92/$05.00
DMS-8905615
and DMS-9007986.
0 1992 - Elsevier Science Publishers
B.V. All rights reserved
29
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2 January 1992
smoother. Our Lemma 5.2 below is also of independent interest, see condition A5 in Mallows (1980, p. 701). To give the formal definition of the 3R smoother, consider the running median mapping R : RE ---)Iw‘, defined for x = (x~)~ E E by R(x) ByRN=Ro
= (med(-+,,
xk?
Xk+l))k~E~
0 R we denote the n th iteration X0 is defined as
. . .
with the input
X” = lim.,,
(1)
(composition)
of R. The output
Xm of the 3R smoother
R"(X')
(2)
provided the limit exists (on each coordinate). From discussion in Tukey (1977), Mallows (1979) and from Theorem B in Bradley (1984) it follows that the limit (2) exists a.s. and is stationary and mixing, provided the input X0 = (X,“) is strictly stationary and mixing in the ergodic theoretic sense. Notice that when applied to finite sequences of data, the algorithm is usally modified by discarding end-point observations in each iteration, see e.g. Tukey (1977, pp. 210-213). This does not lead to significant loss of information, as the algorithm usually reaches fixed point in several iterations; the last claim is supported both by the numerical evidence on several sets of real data in Tukey (1977) and by theoretical estimates (3.1) and (3.2) below. It might be also worth pointing out that the 3R smoother algorithm with an i.i.d. input is an example of the ‘cellular automaton’, see e.g. Schonmann (1990) for another ‘transition rule’ and references. Except in the simplest case d = 1, little seems to be know about asymptotic behavior of variants of the 3R smoother defined for general Zd-indexed (even O-l valued) i.i.d. input sequences. We shall be interested in the strong mixing properties of the output X”. Those will be expressed in terms of the strong mixing coefficients defined as follows: be a strictly stationary sequence of random variables. Define past and future u-fields Let x=(xk>kGz 9,(X) = u( X1 : j < k) and pn( X) = a( X, : j > n) (the dependence of u-fields on X will be omitted, if no misunderstanding results). Put CX,(X)=SU~{(P(A~B)-P(A)P(B)):AEP~(X), (P,(x)=sup{IP(BIA)-P(B)I:AE~o(x),
4??(X)= sup
P(A n B)
il
P(A)P(B)
- 1 : A Ego(X),
BEF~(X)}, BE%(X),
BEE(X),
(3)
%+O},
P(A)P(B)
(4)
> 0 ) 1
i,J where the supremum in (6) is taken over all finite PO(X)-measurable partitions {A;} and over all finite sn (X)-measurable partitions { B, }. A stationary sequence X = (Xk) is called: - strongly mixing, if a,(X) -0 as n -+ cc; - +-mixing, if +n(X) + 0 as n ---f co; - +mixing, if q,(X) + 0 as n --j co; - absolutely regular, if p,(X) + 0 as n + 00. It is known that 0 G a,(X) G /I,(X) G I&(X) G 4,(X). Dependence of left hand sides of (3)-(6) on X will be omitted, if no misunderstanding results. ‘ e as m + 00’ instead of O(m); w denotes the standard Brownian In this paper we use notation motion on [O,l], [x] denotes the integer part of x.
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2. Results Theorem 1. Suppose that X is a stationary absolutely regular randon sequence. Furthermore there is 0 < q < 1 and C > 0 such that for all n, k >, 1, P&k+4(Xa))
G P,,,(X)
+ CW
Then X” = Rm(X)
exists.
” %1(X)).
(7)
(Our proof gives C = 4(1 - q)-‘,) The following corollary lists some of the consequences of Theorem 1; in particular, it shows that the absolute regularity is preserved by the 3R smoother and explicit bounds for the rate of convergence to 0 are found. Corollary (i) Zf (ii) Zf (iii) If n -+ co. (iv) If
1. If X p,,(X) p,(X) p,,(X)
is + < <
a strictly stationary random sequence then the following implications hold: 0 as n -+ co, then X” = R”(X) is defined a.s. and &(Xm) + 0 as n ---) 00. ne8 as n + 00 for some 8 > 0, then /3,(X”) -=z nPej2 asn-+co. pn as n + co for some 0 < p < 1 then there is 0 < q < 1 such that /3,(X”) < qfi
X is m-dependent,
then j3,(Xcc)
Since it is known that aperiodic Harris (1986), we get also the following result.
as
+z p” for some 0 -=zp < 1. recurrent
Corollary 2. If X = (X,), rZ is an aperiodic X” = R”(X) exists and is absolutely regular.
Markov
Harris
chains
recurrent
are absolutely
stationary
Markov
regular,
see e.g. Bradley
chain,
then the output
q
From Corollary 1, using limit theorems known in the literature, one can deduce limit theorems for partial sums of X” under appropriate assumptions on a rate of convergence to 0 of b,(X) and under some additional moment assumptions. However, if the input sequence is $-mixing, we do not need to impose any conditions on the rate of convergence of Q,,(X) to 0. This is achieved by using the approach developed in Bradley and Peligrad (1986), together with the following result. Theorem 2. Suppose that X is a strictly stationary a.s. and there are C < co, 0 6 p < 1 such that IP(A for allA
f-l B) - P(AP(B)
ELP,,(X~),
IG
G-mixing
~[.,2-4](X)f$4)
random sequence.
+
CP”
is defined
(8)
BEAM.
The following result establishes the Central Limit Theorem sequence; no mixing rate assumption is required. Let ST=
Then X* = R*(X)
i:
xr,
under
G-mixing
dependence
of the input
u,’ = Var( Snm).
k=O
Theorem 3. Let X be a strictly stationary $-mixing random sequence. Suppose E{ 1X,, ) 2ts} -C co for some 6 > 0. Then X” = R”(X) exists, and E{ 1X,M ) ‘+‘} < co. If in addition for each E > 0, u,‘/n’-’
+ 00
(9) 31
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STATISTICS & PROBABILITY LE’M’ERS
as n + CO. then (Snw - nE{ Xr })/a,, + N(0, 1) in distribution
2 January 1992
as n + 00. Moreover,
(10) weakly in D[O,l] as n + co. Furthermore, if X is symmetric then E{ XF} = 0.
(i.e., ( XP)P E z and ( - XP)P E z have same finite dimensional
distributions),
The following theorem shows that if the input is an i.i.d. sequence, the moment assumptions for the Central Limit Theorem can be significantly relaxed. In particular the CLT holds even when the input sequence does not have finite second moments. Furthermore, the rate of convergence in the Central Limit Theorem can be analyzed, and (9) can be dropped. Theorem 4. Suppose X is an i.i.d. non-degenerate random sequence such that E{ 1X0 )log’ +&12(1 + 1X0 I)} < cc for some 6 > 0. Then X” = R”(X) exists and E{ ] X7 I2 log2+“(1 + ( X,M I)} < co. Furthermore o,‘/n --$ a2 > 0 as n + M and (S,m - nE { Xr })/an’12 -+ N(0, 1) in distribution as n + 00. Suppose in addition that E{ 1X,, ( 3/2 } < 00. Then there is a constant B = B(X) such that < Bn-7’2
-CX Moreover,
if X,-, is symmetric,
then XF IS symmetric
log’ n.
and hence E { XT}
Remark. If for some 0 the distribution of X,, - 8 is symmetric, symmetric; in particular Theorem 4 allows to write the normal has no second moments. Theorems 1, 2 and the corollaries and 5 respectively.
are proved
in Section
= 0.
then the same proof shows that XF - 13is confidence interval for 8 even when X0 - 8
3. Theorems
3 and 4 are proved
in Sections
4
3. Proofs of Theorems 1 and 2 Our proof monotonic
follows the general idea of Bradley (1984). An ordered triplet of numbers if either a 6 b < c or c < b < a holds. Define the ‘random times’: T+=inf{t>,O:(X,_,, T-=inf{s>O:(X_,_,,
X, , X, + , ) is monotonic} X_,,
X_,+,)
(a,
b, c) is called
,
ismonotonic)
We shall show that that for every n, m, k >, 1, P(T”
>, n + m + k) < P(T”
>, n)P(TO
> m) + CY~(X),
P(T”
>, n + m + k) < P(T”
>, n)P(T”
> m) + $,(X)P(T”
(11) >, n),
(12)
where To is either T+ or T-. To prove (11) and (12) we shall consider the case To = T+. Since T- is the same as Tt for the time-reversed sequence, the proof for the case To = TP is similar and is omitted.
magnitude. In particular Observe that Tf > n means that the numbers X0, X,, . . . , X, _ 1 are of alternating and T+ > n + m + k implies the occurrence of both events A = { T+ > n } and T+>, n is .%$_, -measureable 32
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Volume 13, Number 1
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LETTERS
X,, X,,,) for any n + k G r Q n + m + k}. Hence P(T” B = {there is no monotonic triple (X,-i, using (3) or (4) respectively we get + k) G P(A n B). Since A Ed,_, and B E.S$+~-,(X),
>, n + m
P(AnB)~P(A)P(B)+cu,(X), P(A
n B) G P(A)P(B)
+G,(X)P(A).
By stationarity P(B) = P(T o > m). Therefore (11) and (12) follow. We shall also need the following observation: If X is strictly stationary and mixing (in the ergodic theoretic sense), then P(( X_,, X,,, X,) is monotonic) > 0, see Bradley (1984, p. 254). In particular P(T” an) G P(T” 2 1) = 1 - P((X_,, X0, X,) is monotonic) -C 1 for every n 2 1. Put q = 1 - P(( X_,, The following
X0, X,) is monotonic).
two lemmas
(13)
make use of (11) and (12) to estimate
tails of random
variables
T+ and T-.
Lemma 3.1. For every n, k >, i, P(T”
> nk) < 2(1-
q)-‘(a,_,
V
(14)
q”),
where 0 < q < 1 is defined by (13) and cxk = CQ(X)
is defined by (3).
Proof. Since P(T o 2 k) < q, from (11) we get P(T”
>, (a + 1)k)
> nk) + CQ_,.
Therefore P(T”&nk)
...
<2(1-q)-1(q”vcu,_,).
+q(Cy,_,+q2)...)=qn+cx_l
‘;g;’
0
Lemma 3.2. If $” + 0, then there are 0 Q fi < 1 and d < cc such that P(TO>n)<@
foreueryn>l.
Proof. Let k 2 1 be such that 0 = c@~_,+ q < 1, where q is defined P(T o 2 (n + 1)k) G 0P(T o > nk). Hence P(T o > nk) 6 8” and the
C=eP.
by (13). Inequality (12) implies conclusion holds with F = 61’k,
0
We shall also use the following Lemma A. ZfrfA~a(R~(X),:O
B
lE+,v+i
of Bradley
(1984) Lemma
j< -n), thenAn{T-
The proof of both Theorems A EEn-,?-,
consequence
1 and 2 is now concluded
(X”)ca(Rk(X),:j<
as follows:
2.4.
0 Fix N, k, n > 1 and let
-n-N-l),
(X”)~cr(R~(X),:j>n+N+l). 33
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1992
Define TN =inf{t>O:(X,+,_,,
XrfN,
T; =inf{t>O:(X_,_._,,
X,,,,,)
Xmr-N,
ismonotonic},
XPr-N+,
) is monotonic}.
Clearly TN+ and T; satisfy the conclusions of Lemma 3.1 and Lemma 3.2. Indeed, and T- respectively for shifted stationary sequences (Y,), t L = (X, * N)k E E. We have
they are equal to T+
{T;~n}nBn{T,i~n})
P(An
~P(AnB)~P(An{T~~n}f3BB{T~fn})+P(max{T~,TT,-}>n).
(15)
By Lemma A applied to the shifted sequences (Y,), E z = (X, * N)k E =, we have P_,+,(X) and B n {T,,, G rz}E.F&~(X).Therefore
n {T;GH} nsn
P(A
A
n {T; G n } E
{TN+
G'(A~{T,-Q~})P(B~{T,+~})+~,,~,(X)P(A~{T,-~~}) ~fyA)~(B)+(P,,-,~(A). Lemma
3.2. together
with (15), (16) gives therefore
nB)-P(A)P(B)<+,,_,P(A)+
P(A
(16)
2&n+‘.
(17)
Since for a complement B' of B we have P(A nB')- P(A)P(B')=P(A)P(B)- P(A nB), inequality (8) follows for n = 4k + 2 from (17) by picking N = n = k in (17). The general case then follows by the standard argument, yielding C = 2@j2 and p = $/4. 0 Similarly we prove Theorem 1. Let { Ai) be a R n _ v_ 1(X*)-measurable partition. Then F n+ N+, (X”)-measurable ;C/P(AinB,) i,j
partition
and let { B, > be an
-%4)+,)/
<:CIP(A,n
{T;
nB,n {T~~n})-P(A,n{T',
i.i ++xP(A,nB,n{T,f i,i
>n})+:~P(A,nB,n{T,->n}) i,i
+iCP(A,)f'(B,n(T,t >n})+;~P(A,n{T,-w})P(B,) i,i i,i
nk for n and putting P 4&+4(Xm)
GP,,,(X)
N = nk + 1 we get
+40 -~)-l(%-lVcf)~
q
which is a Proof of Corollary 1. All cases except (iv) follow from /34n~+4(X”) G (C + 1)q” V /3,-l(X), trivial consequence of (7) applied with k = n. To obtain (iv) we choose k > 1 such that cy,_,(X) = 0. Then for n 2 1, Dznk = 0 and by (7) we have P 4n/c+4 34
G
cc
+
lh”.
0
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LETTERS
1992
4. Proof of Theorem 3 The following
simple observation
med( -a,
-b,
-c)
is a consequence = -med(a,
Lemma 4.1. Zf R”( X) exists, then R”( -X) 0 is a symmetric random variable. The next observation
of the fact that
b, c). = -R”(X).
shows that the 3R smoother
Lemma 4.2. Zf X is a stationary
In particular,
preserves
4.1, inequality
P((X,“12t)63P((XO(&t)
(18)
(18) implies
XF > t implies
that at least one of the variables
also that P( Xr < t) < 3P( X0 < t) for every t E R;
foreveryt20.
(19)
Proof of Theorem 3. We shall apply Theorem 1 in Bradley and Peligrad (1986) By (8), condition (1.5) of Bradley and Peligrad (1986) is satisfied (X?)kEZ. geometrically. Indeed for every A E .FY,,(Xco), B E F2,+ s( Xm), n@
-P(A)P@)J
G /p(B)/\&4 < /P(B)
then XT
exists then
foreveryttE.
Proof. Indeed, it is enough to observe that the inequality X _,, X0, Xi cannot be smaller than t. 0
IP(A
in distribution
the moments.
sequence of random variables such that R”(X)
P(X,“>t)<3P(X,>t)
Remark. In view of Lemma in particular
if X G -X
n@
to stationary sequence with {a,} decreasing
-P(A)P@)\
+,,J’( A) + Cp2n G &&GQ~~
+ GP”.
By (19) we have
E{ 1x7 12+6} = (2 + fi)/atl+aP(
1X,m ) > t) dt < 3E( ) X,, )*+a} < CO.
0
Since by assumption 0,:/n' -’ + co as n ---*CO for every E > 0, (10) follows from the conclusion and Peligrad (1986). If in addition X0 is symmetric, then E{ XT} = 0 by Lemma 4.1. q
of Bradley
5. Proof of Theorem 4 Recall that the sequence (X,), E z is associated, see Esary, Proschan and Walkup (1967) if for each m 2 1 and every bounded continuous coordinatewise nondecreasing functions f, g : R*“‘+’ + Iw we have Cov(f(X-, ,..., X,), 8(X_, ,..., X,)) >, 0. Lemma 5.1. Zf X is i.i.d.,
then R”(X)
exists and is associated.
Proof. Indeed, X (being i.i.d.) is associated and since x, y, z + med(x, y, z) is the nondecreasing function of each coordinate, therefore R”(X) is associated for each n 2 1. The limit R”(X) exists because 35
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of the i.i.d. assumption 0
& PROBABILITY
and hence is also associated,
2 January
1992
(1967) property
P5.
LETTERS
see Esary, Proschan
and Walkup
Remark. The proof of the lemma shows that if X is associated, then so is X”. In particular 2 0 for every k 2 1 and lim inf, _ ,n-’ Var(C, d n XT) > 0 in the nondegenerate case.
Cov( X7,
XF)
Lemma 5.2. Suppose f ( .) 2 0 is a nondecreasing and differentiable function, f(x) Z 0 for x f 0 and X is an i.i.d. sequence such that E { f ( l X0 1)) < co. Then the output X” = R”(X) satisfies E{ f2( I Xl
I>} < CQ.
(20)
Proof. Let (XL),,, = R(X). Since XA > t implies that at least two of the variables therefore by the i.i.d. assumption P(Xi 2 t) < 3P2(X0 > t). Lemma 4.1 gives
X_,,
X0, Xi exceed t,
P( I x; (2 t) < 3p2( I x, I 2 1) for each t 2 0. This, together
(21)
with (19) implies
P( 1x,m I > t) f 9P2( I x, ) > t). Since by Chebyshev’s known tail integration
inequality formula:
(22)
P( l X,, l > t) G E{ f( l X0 ])}/f(t),
(20) follows
from (22) and the well
sc+18E{f(lX,l)}~mf~(t)~(lXol~t)dt~~.
Remark. Inequality
(22) can be complemented
P( 1x; 1 > t) ,, Indeed
XT
:P’( 1x,
•I
with the lower bound
1 > t).
> t holds on { X0 > t } f~ { X, > t } and Lemma
(23) 4.1 ends the proof.
Remark. The constant in (22) can be improved to 9 by using Mallows (1979) formula for the distribution of X$‘. Lemma 5.2 holds also for +mixing input sequences X which satisfy q,(X) < cc (with the constant in (21) replaced by 3(1 + $i(X))). The condition #,(X) < cc is satisfied in particular for Markov chains on ([E, .G@)with one step transition probabilities P(x, A) such that there is C < cc satisfying P(x, A) G CP( y, A) Vx, y E E, VA E .?8 (such Markov chains were studied e.g. in large deviation theory). We shall also need the following Lemma 5.3. Suppose c%(Y)
Y = (Y,)
= P”
E{ q2 log’+‘(l
lemma.
is strictly stationary,
forsome
O
+ I r, I)} < co.
Assume also E{ Y,} = 0 and put S,, = Ck <“Yk, u,’ = Var(S,). Then n-’ Moreover, if CJf 0, then S[nt]/u, + w weakly in D[O,l] as n + 00. 36
(24) (25) a~*+u2forsome
a’>,0
asn+oo.
Volume
13, Number
1
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& PROBABILITY
LETTERS
2 January
Proof. First by Lemma 1 in Herrndorf (1985) specialized to the function G(x) = x*(log(x + l))‘+‘, S > 0, and with f,(x) = 2i+’ llog x 1-u+‘) as x + 0, x > 0, we can find a constant C such that IE{Y,~})I~Clloga,I-(‘+“)(E{Yo’ Therefore
log’+s~l+Y,l})2
the sum u2 = E{ Yi} + 2 CyZ1 E{ Y,?} Var
=nE{Y;}
+2
5
converges.
(n-j)E{Y,Y,}
1992
where
foreveryial.
Now by stationarity =no*(l
+0(l))
asn+cc,
j=l
which proves the first part of the lemma. The second part is now a particular after Theorem 2 in Herrndorf (1985). 0
case of the corollary,
stated
Proof of Theorem 4. In order to prove the first part of Theorem 4, we have only to verify the assumptions of Lemma 5.3. As follows from Theorem 1 (or Bradley, 1984, Corollary l(iv)), (XF), =z is stationary and strongly mixing with (Y, < p” for some 0 < p < 1, which proves (24). By Lemma 5.2 applied with f(x) = x log’ +&‘*(1 + x), condition (25) is satisfied. To verify that u f 0, notice that by Lemma 5.1 (XF) is an associated sequence and in particular lim inf, _ m Var(E.jl= i X:)/n > Var( XF). From the lower bound (23) we have Var( XT) # 0, provided X0 is not a constant. Therefore u Z 0 and the first part of Theorem 4 follows now from Lemma 5.3. The second part of Theorem 4 follows from Theorem 2.1 in Birkel (1988) applied to stationary sequence ( Xp)k E z. Indeed, (Xp) is associated by Lemma 5.1. Moreover, by Lemma 5.2 we have E{ 1XT 13} < cc; therefore assumption (2.3) in Birkel (1988) is satisfied. Since it is well known that Cov(X,“, X;) < [a,JX”)J”3(E{ I X,m 13}>*I3 by Theorem 1 (or Corollary l(iv) in Bradley, 1984), condition (2.1) in Birkel (1988) is satisfied. Therefore the conclusion of Theorem 2.1 in Birkel (1988) ends the proof of Theorem 4. q
References Birkel, T. (1988) On the convergence rate in the Central Limit Theorem for associated processes, Ann. Probab. 16, 16851698. Bradley, R.C. (1984), Some mixing properties of Tukey’s 3R smoother, Stochastics 11, 249-264. Bradley, R.C. (1986) Basic properties of strong mixing conditions, in: E. Eberlein and M.S. Taqqu, eds., Dependence in Probability and Statistics (Birkhauser, Boston, MA). Bradley, R.C. and M. Peligrad (1986) Invariance principle under a two part mixing assumption, Stochastic Process Appl. 22, 211-289. Esary, J.D., F. Proschan and D.W. Walkup (1967), Association of random variables with applications, Ann. Math. Statist. 38, 1466-1474.
Hermdorf, N. (1985), A functional central limit theorem for strongly mixing sequences of random variables, Z. Wahrsch. Verw. Gebiete 69, 541-550. Mallows, C.L. (1979), Some theoretical results on Tukey’s 3R smoother, in: T. Gasser and M. Rosenblatt, eds., Smoothing Techniques for Curve Estimation. Lecture Notes in Math. No. 757 (Springer, Berlin). Mallows, CL. (1980) Some theory of non-linear smoothers, Ann. Statist. 8, 695-715. Schonman, R. (1990), Critical points of two dimensional bootstrap percolation like cellular automata, J. Statist. Phys. 58, 1239-1244. Tukey, J.W. (1977) Exploratory Data Analysis (Addison-Wesley, Reading, MA).
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