- Email: [email protected]
Isotonic approximation in Ls
JOURNAL
OF APPROXIMATION
THEORY
Isotonic
31, 199-223 (1981)
Approximation
in L,
DIETER LANDERS Mathematisches Institut der Universit6t .zu K&‘n, Weyertal 86-90, D-5000 K&I 41, West Germany
L. ROGGE Universittit-Gesamthochschule Duisburg, Posflach 10629, Fachbereich IlfMathematik, 4100 Duisburg I, West Germany Communicated by P. L. Butzer Received October 4, 1979
1. INTRODUCTION AND NOTATION Let (Q, d, P) be a probability space and 0 < s < co. Denote by L&R, M’, P) the system of all equivalence classes of J-measurable functions X: O-+ IR with IlXll, := [l lXISdP]“” < co. For s > 1 the space L,(k4.&‘,P) endowed with the norm 11IISis a uniformly convex Banach space. Let SP c ~8’ be a u-lattice, i.e., a system of sets which is closed under countable unions and intersections and contains 0 and R. A function X: R + R is P-measurable if {X > a} E 9 for all a E R. Denote by L,(y) the system of all equivalence classes in L&2, J, P) containing an ymeasurable function. Then LS(.P) is a closed convex set. Let X E L,(Q, J, P); an element YE t,(P) is called a conditional s-mean of X given the o-lattice 4p, if /IX - Ylls = min(/IX - Z(I, : Z E L,(P)}. For s > 1 the uniform convexity of L, guarantees existence and uniqueness of a conditional s-mean of X given 9, denoted by PYX. The above minimization problem is very important, since it allows one to treat approximation problems under order restrictions. For instance, let P be a probability measure on the Borel-field B” of the n-dimensional Euclidean space R”. Let J be the family of all functions YE L&T?“, IB”, P) which are monotone nondecreasing in each component. Define 4p := {C E [B” : x E C, x ,< y 2 y E C}, where x < y means x, < yi for all components. Then 9 is a 199 0021-9045/81/030199-25502.00/O Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in my form reserved.
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LANDERSANDROGGE
a-lattice and A is the system of all g-measurable YE L,(lR”, iB”, P). Hence for each XE L&R”, iB”, P) the conditional s-mean of X given 9 is the unique function which is monotone nondecreasing in each component such that the ]] (I,-distance from X is minimal among all functions which are monotone non-decreasing in each component. For s = 2, approximation problems of this type are highly relevant to the theory of isotonic regression. Illuminating examples and the broad theory of isotonic regression can be found in the book of Barlow et al. [3]. Although much work has been done for the case s = 2-the statistical inference under order restrictions-little is known for the case s # 2. The concept of conditional s-means given a a-lattice was introduced in Brunk [8]. For s = 2 it is the usual conditional expectation given a u-lattice, defined on L, (see [5]). If L? is a u-field one obtains the s-predictors in the sense of Ando and Amemiya (see [ 1I), which coincide for s = 2 with the conditional expectations. The theory developed here traces back to Kolmogoroff [ 131 and Wiener [20], it is intimately related to Wald’s decision theory and plays an important role in Non-linear Prediction, Filtering, Regression and Bayes Estimation. In a forthcoming paper we apply this theory to define and estimate a “natural” median (i.e., a “reasonable” selection of a median). At first we collect some properties of c on L, and prove an integral inequality for the conditional s-means (Theorem 2.11.) which is essential for the further considerations (see Section 2). In Section 3 we extend the operator c from L, to L,-, as the unique operator preserving the property of monotone continuity. An example shows that even for L/ = (0, Q} a monotone continuous extension to larger spaces L, (r < s - 1) is not possible in nearly all cases. In Section 4 we show that FX convergesP-a.e. to PymX if X E L,-, and the a-lattices 9” increase or decreaseto the a-lattice L& . This result contains and extends other results in this direction. For s = 2 and u-fields it is a classical martingale theorem and yields a result of Brunk and Johansen (91, for s # 2 and a-fields it extends a result of Ando and Amemiya [ 11. In Section 5 we prove maximal inequalities, i.e., inequalities for P{su~,,~ leXl> a] and I su~,,~ ]P,“XI’ dP. Our inequalities applied to s = 2 and u-fields yield the classical maximal inequalities; applied to s # 2 and u-fields they lead to much sharper maximal inequalities than those of Ando and Amemiya (see [ 11). For u-lattices they seemto be the first results in this direction. In Section 6 we start with a characterization result for conditional s-means with respect to u-fields (Theorem 6.1). This result, applied to s = 2, yields the characterization results for classical conditional expectations of Bahadur [2], Douglas [lo], Moy [15] and Pfanzagl [ 171. Then we characterize
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ISOTONIC APPROXIMATION IN L,
conditional s-meanswith respect to a-lattices (Theorem 6.4). For s # 2 this is the first characterization result of conditional s-means given u-lattices. For s = 2 we prove a further characterization result (Theorem 6.3) which contains Dykstra’s [ 1l] characterization for conditional expectations given a u-lattice. 2. PROPERTIES OF Pf’I L,
Let (0, ,oP,P) be a probability space. During this section let 9 c & be a fixed u-lattice and 1 < s < co. The conditional s-mean T = c: L, -+ L, has, according to Brunk [8], the following properties T is idempotent; i.e., T( TX) = TX,
(2.0)
T is positive homogeneous;i.e., T(uX) = aTX,
a >O;
T is translation invariant; i.e., T(X + b) = TX + 6,
(2.1)
b E R,
(2.2)
and fulfills
!-(X-TX)s-ITXdP=O; I (X-TwZdP
(2.3)
if
Z EL,(Y),
(2.4)
where aS-’ = l~(~-‘sign a for a E IR. Relation (2.4) aplied to Z E +l and Z E -1 implies that T is s-expectation invariant; i.e., (X - TX)s-‘dP I
Using that XLTX
> (X - TXF
= 0.
(2.5)
TX with > if TX # 0, we obtain from
(2.3) T is s-strictly monotonic at 0; i.e., xs-L’TXdP>
0
for
(2.6)
TX# 0.
As T = c is a nearest point projection onto a closed convex set of the uniformly convex space L, it is well known that T is continuous in the norm topology of L,.
(2.7)
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Brunk [8] has shown that a
(2.8)
Let X < Y and define Z, = TX A TY, Z, = TX V TY. Applying Lemma 7.2(iii) pointwise to a = Y, b = TY, c = X, d = TX we obtain by integration
Since Z,, Z, E L,(9) we obtain ]]Y - TYI(, = /] Y - Z,ll, Z, = TY, i.e., TX < TY. From (2.7) and (2.8) we obtain that T is monotone continuous; i.e.,
and hence
(2.9)
X, r X (X, 1 X) implies TX, T TX (TX, 1 TX). Let 2 := {c : C E 9} be the dual u-lattice of 9. Since L,(9) = --L,(g) follows directly from the definition of s-means that PTX = -PY(-X),
XE L,.
it
(2.10)
Now we prove an integration inequality which is an important tool in the following sections. For s = 2 this was proven in [3, p. 3421 by other techniques. Denote by [B the Borel-field of I?. THEOREM
Y-measurable.
2.11. Let rp: I? -+ [0, 00) be B-measurable and Z: R -+ R be Then we have for all X E L,
if the integral exists. ProoJ W.1.o.g. we may assume that Z is bounded. By the usual techniques it can be seen that it suffices to prove the assertion for nonnegative bounded functions p with bounded derivative. Let such a q be given and it4 := suplen l@(t)l < co.
(i) We prove the assertion for Z = 1, with C E 9: Let Y,:=~X+(a/M)y,oP~X,OOaa l.Then Y, = ly, 0 PYX,
where y,(t) := t + (a/M) p(t), t E I?, 0 < a < 1.
(1)
ISOTONIC
APPROXIMATION
203
IN L,
Since w,: R -+ R is monotone increasing, we obtain from (1) that Y, is y-measurable for all 0 < a < 1.
(2)
Define Z, :=PyX+
(a/kf)l,q70
PYX,
O
1.
Since q~> 0 we obtain with (2) for all b E R, 0 < a ( 1 {Z,>b}={I$%>b}U(Cn{Y,>b})W and hence Z, is y-measurable for all 0 < a ( 1.
(3)
The function da) := IM- Z&9
O,
attains its minimum at a = 0, because Z, = P,“X and Z, EL,(Y) a E [0, 1). Hence O
for all
[(X-P~X)~l,a,oP~XdP,
which implies the assertion for Z = 1,. (ii) The assertion holds for all non-negative bounded Y-measurable functions by (i), since every non-negative Y-measurable function is the monotone limit of non-negative y-measurable simple functions. (iii) Now let Z be p-measurable and bounded. Since Z = Zt - Z-, it suffices, according to (ii), to prove that -
i
(X - P:XFZ-q
0 PYX dP & 0.
(4)
The function I&) := rp(--t), t E R, is a non-negative bounded B-measurable function. Hence (4) is fulfilled if we show i
(-X - (-cX)FZ-
I//(-P,“x> dP & 0.
(5)
Let 2 = {C: C E 9} be the dual lattice of 5R. Since Z- is g-measurable and -PyX = PT(-X) by Property 2.10, (ii) applied to the u-lattice @ yields (5).
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3. EXTENSION AND PROPERTIES OF FILL,-,
Up to now the operator Py is defined on the domain L,. For s = 2 it is known that Pi;” can be extended from L, to L, . It is our aim to extend for general s > 1 the conditional s-mean Py as a monotone continuous operator from L, to larger spacesL,, more exactly to choose r as small as possible. For instance, can we always extend the conditional s-mean from L, as a monotone continuous operator mapping L, into L 1? It turns out that we can extend Py from L, to L,- I but, even in extremely simple cases, not beyond L,- , . This shows, for instance, that for s > 2 we cannot extend to L, but for 1 < s < 2 we can extend even beyond L, . We write Pf’/L in order to indicate that Py is defined on L c L, ~, . DEFINITION 3.1. (i) Let f!-i be the system of all XE LS-,(.R, AZ?‘, P), which are bounded from below by a function of L,. For X E f!- , define
*x
:= ljz P,“(XA n).
We remark that this limit exists since Py/L, is monotone and X A IZ, n E N, is a nondecreasing sequence in L,. It is easy to see (use Property (2.9)) that P,“lZ!,- , is a monotone extension of PYL, . (ii) For X E L,-l define Pf!X := ljz P,“(X V (-n)). We remark that this limit exists since Pfjf?+, is monotone and XV (-n), n E N, is a non-increasing sequencein f?-, . Since it turns out that P,“/L!,-, is monotone continuous (this is proven in Theorem 3.2), PSy’L,- 1 is an extension of c/E, _, and hence of PF/L, . THEOREM 3.2. Let g c A’ be a o-lattice and 1 < s ( co. Then the operator PT/L, _ 1 has thefollowing properties:
(PO) (PI) (P2) (P3) (P4) (P5) (P6)
e maps L,-, into LSpI, pf”IL,-, is idempotent, c/L,-, is montone, c/L, _, is positive homogeneous, c/L,-, is translation invariant, c/L,- 1 is monotone continuous, c/L,-, is s-expectation invariant,
ISOTONIC
APPROXIMATION
205
IN L,
(P7) J(x-~*z~o~XdP
is y-
(P8) I(X-eX)s-IpopYXdP=O measurable and the integral exists.
is B-
if XEL,-,,
(0: R-+R
Proof. We prove the properties above for Py/l!- r, using the corresponding properties of Py/L,. Using the then proven properties of Pf’/Q!,- r, one obtains in a similar way the stated properties of Py/L,-, .
(PO) Let X E I?-, . Then X, := X A IZE L,. As PF/L, is s-expectation invariant (see Property 2.5) we obtain P[(X,-PyXJq
=o
(1)
AS(X,-P~X,~~(X-P~X,)S-‘~~~XE~,_,,P~X,EL,,(~)~~~~~~~ o
Since (X - PyX,p
nEN
< co,
1 (X - PYXF,
(2)
(2) implies
o
< 00
and hence X - PTX E L,-, . As L,-, is a linear space and X E L,-, , consequently PYX E L,- , . Since X> YE L, implies P,“X> PYY, we get PTXE Q!,-,.
Property (Pl) follows directly
from (2.0) and the definition of
Pyp-,.
Property (P2) follows from (2.8) and the definition of PF/lI?S-, . Property (P7) W.1.o.g.we may assumethat Z is bounded. By the usual techniques it can be seen that if suffices to prove the assertion for nonnegative, continuous and bounded functions (o. Let now X E !i!!,-, and X, := X A n. Then X, E L, and therefore Theorem 2.11 implies
I (X, - P:X,pZrp
0 P,“X, dP < 0,
n E N.
As q is continuous, v, o PYX,, -+ p o P,“X by definition of P,“X. PYX E L,- 1 according to (PO), it is easy to see that
Since
is bounded by an integrable function. Hence the Theorem of Lebesgue implies the assertion.
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AND ROGGE
Property (P6) follows directly from (P7). Property (P5) W.1.o.g. we prove only the decreasing case. Let X,, E L,-, with X, 1 XE 2,-r. Then by (P6)
((xn-PyxnpdP=o,
nEN.
(3)
By (P2) we obtain PYX < PTX, ( CX, . Hence the Theorem of Lebesgue implies by (3) s--l
x-
As 1 (X - CXpdP PYX = lim, ENCX”.
p& PYXn
dP=O.
(4)
= 0 and PYX < lim,,, PYX,, , (4) implies that
Properties (P3) and (P4) follow immediately from (P5), using the corresponding properties of P$, (see Section 2). Property (P8) follows from (P7) applied to q’+ and (o- with Z E 1 and Zr-1. The preceding theorem shows that there exists a monotone continuous extension of Py/L, which maps L,- i into L,- r . The following remark shows that, even for 9 = (0, a}, such an extension is not possible for any larger space L, (r < s - 1). Remark 3.3. Let (0, d, P) be a probability space and 1 < s < co. Assume that L,#L,-, for some OO with X E L, -L,-, be given and choose 9 = (0, n). We shall show that each monotone continuous extension of Py/L, to L, necessarily maps X to the function =co &L,.
Let X,, :=X A n E L,. Then X,, T X. If there exists a monotone continuous extension from c/L, to L,, then c, = P,“x T PFX = c. Assume that c < co. Since c/L, is s-expectation invariant we have 0=1(X,,-c,FdP>j+(X,-cFdP>-co.
Since X, T X, this implies 0 > J”(X - c)” dP > -a~. Hence X E L,- , , contradicting our assumption. For each X E L,-, we give now a characterization of the conditional smean eX as the unique g-measurable function fulfilling two integralinequalities. For the case s = 2 this is known for X E L, (see [3]); it seems to be unknown even for s = 2 if X E L, .
ISOTONICAPPROXIMATION IN L,
207
THEOREM3.4. Let X E L,- 1 and cpbe a strictly increasing function such that (Do PYX is bounded. Then Y = P,“X if and only if YE L,- ,(9), rp 0 Y is bounded and (i) (ii)
1(X--YF Z dP < 0 for all bounded 4P-measurable Z; 1(X-- YFqo YdP=O.
Proof. The direction “only if’ follows from (P7) and (P8) of Theorem 3.2. For the converse direction it suffices to prove that if Y,, Y, E L,_ ,(9) are two functions fulfilling (i) and (ii), then Y, = Y,. By (ii) we have
I (x-
Yj)+p
o Y,dP=O,
j= 1,2.
Since q o Yj, j = 1, 2 are bounded y-measurable functions we obtain, together with (i), f ((YI - XF
+ (X - &j+(v)
0 Y, - rp 0 Y2) dP < 0.
Since rp is increasing, it is easy to see that ((Yl - xp
+ (X - Y*)s--I)((o 0 Y, - (00 Y*) > 0.
Hence rpo Y, = rpo Y, P-a.e., whence Y, = Y2 P-a.e. COROLLARY3.5. Let YE L,-,(y)
(i) (4
X E L,-, .
Then
Y = PFX
if
and
only
if
and I,(X-Y)“-‘ldP
YFdP=Ofor
all aE R.
Proof. The direction “only if’ follows from (P7) and (P8) of Theorem 3.2. For the converse direction we have to prove (i) and (ii) of Theorem 3.4. Since v(B) := J (X- wl, 0 Y dP, B E [B, is a signed measure which is zero on the system {[a, co), a E IR} according to (ii), it is zero on IB also. This implies (ii) of Theorem 3.4 even for all measurable functions cp,for which the integral in (ii) exists. Since every non-negative bounded 9measurable function is the monotone limit of non-negative Y-measurable simple functions, (i) implies, that condition (i) of Theorem 3.4 holds for all non-negative, bounded, y-measurable functions Z. Let Z be a bounded 9-
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AND ROGGE
measurable function. Then Z + a is a non-negative bounded p-measurable function for suitable a and hence
i
(X- Y,P(Z+a)dP~O.
Consequently we obtain (i) of Theorem 3.4. Now we give some further properties of the extended Py. Since we obtain from Property (P7) of Theorem 3.2 with L,(-q = -L,(@, Z=-1, that for XEL,-, (P9) j”,,(X-cXpdP>O if DEg, CE(PyX))-‘IB, where I3 denotes the Borel-field of R. Using the monotone continuity of Py/L,+ 1, Property 2.10 implies (PIO) PYX= -PY(-X), XE L,-,. Since X E L,- I and Y = PYX fulfill (i) and (ii) of Corollary 3.5, we obtain for each u < 1 that X0 =X - uP:X and the y-measurable function Y, = (1 - a) eX fulfill (i) and (ii), too. Hence Corollary 3.5 implies: (Pll) Py(X-uP:X)=(l-u)PyX, a,< l,XEL,_,. Since Z$? is monotone we have P,“(--1x1) ,< PyX< P,“lXl. Hence (PlO) implies (P12) IT4 < max(C(Xl, FIXI), XE L,-, . We remark that for o-lattices it is not true that lcX1 (P,“lXl; this can be seen by simple examples even in the case s = 2. Using (P2) and (P8) it is easy to see that for each interval Zc IR (P 13) X E Z P-a.e. implies P,“X E I P-a.e. Now we prove a convexity inequality for conditional s-means. (P 14) Let Z c R be an interval and q: Z + R be a non-decreasing continuous and convex function. If X, q 0 XE L,-, and XE Z P-a.e. then a, 0 PfcX < PY($o 0 X). ProoJ
Since ~1is continuous and convex there exist a,, b, E R, n E Id,
such that
P(X) = “,t~ (w + 4)
for all
x E I.
Since v, is non-decreasing, w.1.o.g.a, > 0. From (P2), (P3), (P4) and (P13) we obtain P,“(u, o x) > sup u,P:X nefh
+ b, = q 0 PYX.
Q.E.D.
ISOTONIC APPROXIMATION IN L,
209
Finally we prove for each r > s - 1 (P15) XEL,=+-PYXEL,. According to (P12), we may w.1.o.g. assume that X > 0. Since Xr’(‘-‘) E L,-, , (PO) implies Py(XV”-“) E L,-, . As, by (P14) (PYX)rlcs-l) < P$‘(X”‘“~“) E L,- i, we obtain PYX E L,. s 4. CONVERGENCE A.E.OF PFX In this section we give a general a.e. convergence theorem for conditional s-means P,‘;lX for all X E L,- i . For the case s = 2, it is a result of [9]. For the special case of o-fields and for X E L, this was proven in [l]. The techniques used here, are related to the methods of Sparre et al. [ 181 and Brunk and Johansen [9]. If Yn c d, n E N U {co }, are a-lattices, Yn, n E iN, increases [decreases] to pa if 9” c ik7,+,[pn 2 49,+i] and pa is the o-lattice generated by
unsN zwm= nnd3
THEOREM 4.1. Let (Q, z.4, P) be a probability space and 1 < s < 00. Let Yn c &‘, n E IN, be o-lattices increasing or decreasing to the o-lattice pm. Then
P-a.e.,
PFX + P,“,X, forallXEL,-,(&sd,P).
Proof. We prove only the decreasing case, which is somewhat more complicated. The proof for the increasing case runs similarly. LetX,=~X,nE~:=NU{co}.Let~~:={C:CE~,}.Accordingto properties (P7) and (P9) of Section 3, we have (X-apdP
if
aElI?,C,EY~,nER,
(1)
and
Define X= bneN X, and x= hm,,, X,; then & X are IP,-measurable. Let C E Ym and a E IF?be fixed and choose a sequence E, 1 0. Define for mQn
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LANDERS
AND ROGGE
According to (I), we have
Ic!r (X-(a+c,)FdP
(3)
Since CL=, Cr,, = en {infmGiGrXi ( a + em}=: Cr and CT t,+, C” := C {inf,>, Xi < a + cm} we obtain, from (3),
n
J (X -
(a + E,))~
m E N.
dP ,< 0,
(4)
Cm
Since lc,+
msN
1Cn{g(at'
(4) implies, using the Theorem of Lebesgue
Jix
if
uE IR,CEYa.
(5)
if
bE I?,DEgm.
(6)
In the same way we obtain (X-bFdP>O, J(W>6jfTD
Now we show, using the relations (l), (2), (5), (6) that RX,,
X,
P-a.e.
This directly implies the assertion. We prove only X,
has P-measure zero, Relation (2) applied to n = co and D, = {X < a} E .@ff yields
J(X-bpdP>O.
(7)
E
Relation (5), applied to C = {X, > b} E ip, yields
JE (X--aFdP
> (X - b)“L on E, (7) and (8) imply P(E) = 0.
04)
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ISOTONIC APPROXIMATION IN L,
5. MAXIMALINEQUALITIES
AND NORMCONVERGENCE
OF P:X
In this section we prove two maximal inequalities and give an application to norm convergence. THEOREM 5.1. Let (SJ,&, P) be a probability space and 1 < s < 00. Let 9” c ,?p, n E N, be an increasing or decreasing sequence of o-lattices. For non-negative X E L,- , we have
ProoJ:
Let X,, = FX,
n E N. It s&ices to prove for each n E N that
As for decreasing sequences9,) n E N, the finite sequencepn, Yn-, ,..., 9, is increasing; it suffices to prove (1) for the increasing case. At first we show that
1
(X-aFdP>O,
a > 0.
(2)
ImaXlal
Let Ai = {X,a}, i=l,..., n. SinceAi={Xi>a}fTDi, where Di E @ we obtain from Property (P9) of Section 3 j
(X-apdP>j A,
(X-X,)“L’dPaO. Ai
As CrrIA, = (max , GiGnXi > a], this implies (2). Applying Lemma 7.2(i) and (ii) pointwise to x =X(w) for all o E {maxICiGn Xi > a}, we obtain by integration, using (2), inequality (1) COROLLARY 5.2. Let (l2, s’, P) be a probability space, 1 < s < 00 and r > s - 1. Let Yn c J/, n E IN, be an increasing or decreasing sequence of alattices. Then we have for XE L, that
where u,,, = (2”-*I [r/(r - s + l)])“‘“-” X is non-negative and 6 = 2 elsewhere.
and 6 = 1 ifall Yn are o-fields or if
212
LANDERS AND ROGGE
ProoJ First we prove the assertion for X> 0. We use the following well known lemma [ 12, p. 23 11: If <, q: R -+ [0, co] are d-measurable with
P{r > s} < (l/a)j
CdP
lV>E)
for all
E > 0,
(“1
then P[ya] < [a/(a - l)]” P[ca] for all a > 1. Let q := (supnENPpX)‘-‘, r= 2ls-21~s-’ ) &=a$-’ and a = r/(s - 1) > 1. Then (*) is fulfilled according to Theorem 5.1. Hence the lemma cited above implies the assertion for X> 0. Hence the assertion is true for o-fields and arbitrary XE L,, using ]eX]
We remark that for the case s = 2 and a-fields we obtain in Theorem 5.1 and Corollary 5.2 exactly the classical maximal inequalities with the same bounding constants. For a-fields 9n and X E L, Ando and Amemiya [l] proved that P
[
“,llR lPs”“Xl”
1
The constants a, given by Ando and Amemiya are much larger than our constants u,,, specialized to the case r = s. For s = 2 we have u,,, = 4 and a, = 208 and for s > 2 there holds (2szQ-
l < u,.
Observe that 1 < u,,, = (21s-2’s)s’(s-‘) + co with s+ co. For u-lattices the preceding results of this section seemto be the first in this direction. COROLLARY 5.3. Let (f2, &, P) be a probability space, 1 < s ( co and r > s - 1. Let Yn c H’, n E N, be a sequence of a-lattices increasing or decreasing to the a-lattice ip, . Then for each X E L, we have
Proof. According to Theorem 4.1 and Corollary 5.2 the assertion follows for r > s - 1 by the Theorem of Lebesgue. For r = s - 1 it suffices, according to Theorem 4.1, to prove that JPSyXjSml, n E N, is uniformly integrable. Since IP?XJS-’ < (PF IXJ)s-’ + (PF (XI)‘-’ and the sum of two uniformly integrable sequences is uniformly integrable (use e.g. [4,
213
ISOTONIC APPROXIMATION IN J!,,
Korollar 20.3, p. 901) we may w.1.o.g. assume that X > 0. Let X, = P%X, n E N. We obtain from Lemma 7.2(i) and (ii), applied pointwise for w E {X, > a} to a =X,(U), x = X(o)-since J,X,>aj (X- X,pdP = 0 according to (PS) of Section 3-that J’lX,>al
xi-’
dP < 21s-*’
I lX,>al
Xs-’ dP,
a > 0.
(+>
Since ~up,,~ P{X, > a} +a+m 0 by Theorem 4.1, we consequently obtain from ( + ) that Xi-‘, n E N, is uniformly integrable. 6. CHARACTERIZATIONS OF Pf’ In this section we characterize the operators Py for u-fields and o-lattices. We start with a characterization result for o-fields and arbitrary s > 1. For s = 2 this leads to a characterization for classical conditional expectation operators which is a common generalization of the results of Bahadur [2], Douglas [lo], Moy [ 151 and Pfanzagl [ 171. THEOREM 6.1. Let 1 < s ( CC and s - 1 < r < co. Let T: L,(Q, J, P) --+ L,(Q, &, P) be an operator which is
(i) (ii) (iii)
homogeneous,
(iv> (v)
monotone,
translation
invariant,
idempotent, s-expectation invariant.
Then there exists a sub-a-field XE L,.
3 c ~2 such that TX =PTX
for all
Proof. Let ,5?:= {A E JY: Tl, = lA}. Since T is homogeneous and translation invariant, A E J implies Tl,= 1 - Tl, = 1 - 1, = 1,. Hence 9 is a a-field according to Lemma 7.3(ii). Since T, e: L, -+ L, are monotone continuous operators according to Lemma 7.3(i) and Property (P5) of Theorem 3.2 it suffices to show that TX= P,“X
for bounded X.
We remark that according to (i), (ii) and (iv) TX is bounded if X is bounded. Applying (iv) of Lemma 7.3 to Z and -Z we obtain
I (X640/31/3-2
TXpZdP=O
for
Z E L,(3).
214
LANDERS AND ROGGE
Hence we can apply Theorem 3.4 with Y = TX and the strictly increasing function q(t) = t, t E R. This yields TX = PFX for bounded X. Easy examples show that none of the five conditions in Theorem 6.1 can be omitted without compensation. Another characterization result for conditional s-means given a a-field can be found in [ 141. In the case s = 2 s-expectation invariance of an operator T means s TX dP = s X dP. Hence for s = 2 monotony and s-expectation-invariance of T trivially imply that T is weak 1)J/,-reducing; i.e., IITX-
TYII, G 11X- YII,
if
X< Y.
Furthermore each linear operator with I]TX/I, < ]lX]l, is trivially weak 1)/Iireducing. Therefore the following corollary contains the characterization results for conditional expectation operators given in [2], [lo], [ 151 and. (171. COROLLARY 6.2. Let 1 < r < co and T: L,(R, ~4, P)-+ L,(sZ, ~4, P) be an operator which is
(i) (ii) (iii) (iv)
homogeneous, translation
invariant,
idempotent, weak 1111,-reducing.
Then there exists a sub-o-field XE L,.
9 c J
such that TX = PfX
for
ail
Proof. According to Lemma 7.4 T is monotone and 2-expectation invariant. Hence Theorem 6.1, applied to s = 2, implies the assertion. The only characterization result, known to the authors, concerning ulattices is the result of Dykstra [ 1l] for the case s = 2. He shows that an operator T: L&Q, d, P) -+ L&2, @‘, P) which is
(i) (ii) (iii)
positive homogeneous, ]I I(,-reducing; i.e., IITX - WI2 < IIX - Yll,, idempotent, (iv> monotone, (v) Z-expectation invariant; i.e., ] TX dP = j X dP, (vi) 2-strictly monotonic at 0; i.e., ] XTX dP > 0 if TX # 0
is the conditional expectation operator c with respect to a suitable u-lattice LFCC. We remark that Dykstra requires instead of (vi) a slightly stronger condition, but he uses in his proof only condition (vi) [3, p. 3221. Dykstra
ISOTONIC
APPROXIMATION
215
IN L,
furthermore gives an example showing that conditions (i)-(v) above are not sufficient for his characterization result [3, p. 3261. In the characterization results for conditional expectation operators given a a-field, only one integration condition besides algebraic conditions is used. This is an advantage since the authors believe that algebraic conditions are nicer and easier to verify than integration conditions. Dykstra’s characterization result for a conditional expectation operator, given a u-lattice, uses three integration conditions. The “strongest” of his integration conditions is the property (ii), namely, 11II,-reducing. In the following theorem we show that 11I/,-reducing can be replaced by the algebraic condition translationinvariance. Furthermore monotony and 2-expectation invariance is weakened to weak /I I/,-reducing. Since 11iI,-reducing and II II,-expectation invariance imply translation invariance (see [3], Proposition 7.2, 2”, p. 324) the theorem below contains the result of Dykstra. THEOREM 6.3. Let 1
T: L,(G!, &, P)+ L,(R, zZ,P)+
(i) positive homogeneous, (ii) translation invariant, (iii) idempotent, (iv) weak I/ /I ,-reducing, (v)
2-strictly monotonic at 0 for XE L,.
Then there exists a o-lattice 9 c J/ such that TX = PyXfor
all X E L,.
Proof. Since according to (iv) T is II iI,-continuous for monotone sequences, it suffices to prove TX = PYX for X E L,. According to Lemma 7.4 the operator is 2-expectation invariant and monotone and hence according to Lemma 7.3 there exists a u-lattice g c & such that
c
(X-TX)ZdP
for X E L,(d),
Z E L,(Y).
According to Theorem 3.4, applied to s = 2, X:=X
(1)
- TX, Y = 0 and
q(t) = t, (1) implies that Pf(X - T(X)) = 0.
(2)
T(X - PFX) = 0
(3)
Furthermore we have
because T(X - KX) E L,(Y) implies l (X - PTX) T(X - Pf”X) dP < 0 and (v) implies I (X - CX) T(X - CX) dP > 0 if T(X - KX) # 0.
216
LANDERSANDROGGE
Now we shall show that
IITXIII= IIe%
XEL,.
(4)
According to Lemma 7.4, the operator T is ]] I),-reducing. Using (2), (3) and the fact that e is ]] III-reducing, too, we obtain:
Il~~lI~=Il~-~~-~~~ll,~ll~~-~~~-~x)lI,=ll~~Il, and
IITXII, =11X-(X-
T’)(l,hll~X-P?(X-
Wll, =llP~Xll,.
Hence (4) follows. Now (4) and translation invariance of T and PT imply ~~--u[~,=l~TX--n~~, for XEL,, aElF?; i.e., ~I~-ulPl(dx)= x - a ] P,(dx) for a E F?, where P, is the distribution of P, X and P, the distribution of TX. Hence according to Lemma 7.5, P, = P, and therefore especially !’ (TX)’ dP = ( (PyX)‘dP.
(5)
Now we obtain by (5) and (I), that 11TX - EXll;
= 2 [ (PFX)’ dP - 2 j” TXPYX dp = 2 =2
XP’;yX dP -
I
i
TXPyXdP
1
(X-TX)PFXdP
and hence TX = PYX. Dykstra’s example cited in [3, p. 3261 (52= { 1,2}, P{ 1) = P(2) = 4, if X(1) > X(2)), TX = X if X(1)
(i) (ii) (iii)
positive homogeneous, translation invariant, idempotent,
ISOTONIC
APPROXIMATION
217
IN L,
(iv)
monotone, (v) s-expectation invariant, (vi) weak s-monotonic at 0; i.e., I Xs-=--‘TXdP > 0 for X E L, TX > 0 and fulfills: (vii) T(X - f TX) = 4TX.
with
Then there exists a a-lattice 9 c JCZsuch that TX = PTXfor all X E L,. Proof. According to Lemma 7.3(ii), (iii) we have that Y := (A E J/: Tl, = 1, } is a u-lattice and TX E L,(Y) as T is idempotent. Since T, Py: L, + L, are monotone continuous operators, it suffices to show that TX = PrX
for bounded X.
For this, it suffices to prove condition (i) and (ii) of Theorem 3.4 applied to the bounded function Y = TX. Condition (i) of Theorem 3.4 is fulfilled according to Lemma 7.3(iv). Since TX is bounded and y-measurable, Condition (ii) of Theorem 3.4 is fulfilled with q(t) = t if we show that i
(X- TX)“TxTXdP>
0.
(*I
Define inductively y, = 1 and yn+r = (1 + y,)/2. We prove by induction T(X-y,TX)=
(1 -r,>TX.
c**>
For n = 1 this is our Assumption (vii). Assume now that (**) holds for n. Then T(X - y,,+ 1TX) = T(X - y,,TX - )T(X - ynTX)) =~T(x-~,TX)=~(~-QTX=(~-~~+,)TX. Hence (**) holds for (n + 1). Let X2 0, then (1 - y,) TX > 0. Hence, according to (vi) and (**), O<
I
(X-y,TXpT(X-yy,TX)dP
i.e., i
(X- y,TXpTXdP>O.
T(X- y,TX) =
218
LANDERS AND ROGGE
As yn--t 1, we obtain (*) for bounded X > 0. As T is translation invariant we obtain (*) for all bounded X. We remark that conditions (i)-(vi) are even in the case s = 2 not sufficient to characterize s-means. This can be seen from the just cited example of Dykstra.
7. AUXILIARY
LEMMATA
At first we prove a lemma characterizing a system of functions as a system of y-measurable functions where 9 is a suitable u-lattice. This lemma plays on analogous role for a-lattices as Lemma 3 in [ 171for u-fields. 7.1. Let 0 ( r < co and 0 # F c L,(f2, -xf, P). Assume that F the following conditions:
LEMMA
jiiljZls
(i) (ii) (iii)
aX+,BEFfirXEF,a>O,/3ElR,
XA Y,XV
YEFforX,YEF,
X,EF,XEL,andX,,fXorX,1XimplyXEF.
Let 9 := {A E ~4: 1, E F}. Then Ip is a o-lattice and F = L,.(p). Proof.
Since lc,n c. = l,, A lcZ, l,,,cl = l,, V lc, and 1n;=,ckl lnkm,,ck~
1u;=,c, T lu:,w
(ii), (iii) directly imply that 9 is a a-lattice. Now we show F c L,(p). X E F be given then for each /I E R
Let
Y,,:= [4X-P) V 01A 11T lfw:xcwj>41. As (i), (ii) imply Y, E F we obtain from (iii) that {o: X(o) > p} E 9; i.e., x E L,(g).
Finally we have to show L,(9) c F. Let X E L,(P). By (i) and (iii) we may w.1.o.g. assume that X > 0. Since X E L,(Y) we have {cu:X(w) > a} E P and hence l,o:XCw)>a,E F. By (i) and (ii) this implies X, := sup $.
l{w:xCw)>+}: v = 0, l,..., n2”
I
As X,, T X, hence X E F by (iii).
I
E F.
ISOTONIC
APPROXIMATION
219
IN J?.,
LEMMA 7.2. Let 1 < s < 00. Then there hold the following between real numbers:
(i) (ii) (iii) Proof
as-’ +2s-2(x-a~<22s-2xs_=1 as-’ +(x-ap<22-Sx”-’
inequalities
ifa>O,s>2, zfa,x>O,
1
la-bVd\“+(c-bAdl”c.
Dividing (i) by as-’ it suffices to prove
f(z) := 1 + 2s-72 - l>sL’ < 2s-3s-1 = g(z),
z E R.
(*)
Since f(l)< g(1) and?(z)< g’(z) f or z > 1, (*) is true for z > 1, s > 2. This directly implies that (*) is also true for z < 0. If 0 < z < 1 use differential calculus to show that h(z) = g(z) -f(z) attains its minimum at z0 = { and h(i) = 0. Inequality (ii) follows in the same manner. Inequality (iii) follows using similar techniques. LEMMA 1.3. Let l
(i) (ii) (iii) (iv)
T is monotone continuous, 9 = {A E JS’: Tl, = lA} is a u-lattice, L,(9)
= {X E L,: TX = X},
J(X-TX~ZdP
Proof (i) We show only that X, T X implies TX, f TX if X,, X E L,. The decreasing case runs similarly. Since T is monotone the pointwise limit Y := lim,,, TX, exists and fulfills Y < TX. As T is s-expectation invariant we obtain
!’(X, -
TX,, p
dP = 0
and hence by the Theorem of Lebesgue
c
(X-
wdP=O.
Together with Y < TX and ( (X - TXpdP Y= TX.
= 0 this implies
220
LANDERS AND ROGGE
For (ii), (iii), let F := {XE L,: TX = X}. Since T is positive homogeneous and translation invariant, condition (i) of Lemma 7.1 is fulfilled. To prove condition (ii) of Lemma 7.1 let X, YE F be given. We show XV YE F, the proof for X A YE F runs similarly. Since T is monotone we have T(XV Y)> TXV TY=XV
Y.
As T is s-expectation invariant we have (XV Y-T(XV
0=
Y)FdP,
whence T(X V Y) =X V Y; i.e., X V YE F. Condition (iii) of Lemma 7.1 follows from the monotone continuity of T. Hence Lemma 7.1 implies (ii), (iii). Now we prove (iv) if 0 =
I I I
(Xl, - T(Xl,))s-r
dP
(Xl, - (TX) lc)ti
dP
1,(X - TX)““L dP.
As T is positive homogeneous and translation invariant we obtain (iv) for all bounded X and Z = l,, C E 9. As T is continuous on monotone sequences we obtain (iv) for all X E Lr(&) and Z = 1, with C E 9. Since every nonnegative y-measurable function is monotone limit of y-measurable simple functions and since T is s-expectation invariant we obtain (iv). LEMMA 7.4. Let 1 < r < co and T: L,(Q, ..M’,P)+ L,(f2, M’, P) be a weak 11[[,-reducing, translation invariant operator with T(0) = 0. Then T is
(9
(ii) (iii)
monotone, 2-expectation invariant, /I ([,-reducing; i.e., ([TX-
TYIJ, < IIX-
YJI,.
ISOTONIC
APPROXIMATION
IN L,
221
Proof. At first we show that T is 2-expectation invariant. Since 7Q = 0 we obtain that
Now let X E L,(jQ) with 0
c~~~ (2)
Hence the two inequalities in (2) are equalities. The first of them implies c = I TX1 + Ic - TX1 and hence 0 < TX < c, therefore the second implies 1‘TXdP=IITXII,=IIXJI,=J-XdP. Since T is translation invariant, 2-expectation invariance holds for all bounded X. Since T is weakly 11III-reducing we obtain that
if X, T X or X, 1 X.
IITX, - T’ll, -+0
(3)
Let X E L,(J) be bounded from below. Then there exist bounded functions X, with X, T X. Since IX,, dP = s TX, dP, we obtain from (3) (jXdP-jTXdP
(=Fg
IjX”dP-jTXdPl
< &$lTX,-TXII,=O.
Hence (X dP = l TX dP for all X E L&f’) which are bounded from below. Since each integrable function is a decreasing limit of integrable functions which are bounded from below, we obtain in a similar manner that s TX dP = j X dP for all X E Lr(~), i.e., (ii). Now we prove that T is monotone. Let X, YE L&t’) with X < Y. Since T is 2-expectation invariant and weak II /,-reducing we obtain j(Y-X)dP=j(TY-TX)dP
222
LANDERSANDROGGE
Hence we have equality, whence TY - TX = 1TY - TX(; i.e., TX < TY. Since T is monotone and weak 11(I,-reducing, T is (1I),-reducing:
LEMMA
1.5.
Let P,, P, be two p-measures on the Borel-field of R. If Ix--alP,(dx)=jIx-alP,(dx)
for all a E R, then P, = P,. Proof It suffices to prove P,(-a~, y] = Pz(--03, y] for all y E R. For all E > 0 we have by assumption
I”x-
YI - lx- (Y + E)l)Pl(dx)= j?lx - YI - Ix--- (Y + c)I)P,(dx);
i.e.,
E[-q--co;Yl + p11.Y+ 6 =)>I+ J (Y.
= E[-P,(-Co,
y] + P,[Y + E, aI1
(Ix-YI-Ix-(Y+~)I)Pl(~) Y+E)
+ J(y,,,+c)(Ix - Yl - Ix - (Y + E)I)P*(dX)*
Divide by E > 0 and let E-+ 0. Then we obtain
-P1(-q u] + P,(Y,a> = -PA-% Yl + P*(Y,03) and hence -2P,(-CO,
y] - 1 = -2P2(-CO, v] - 1;
i.e., our assertion. REFERENCES 1. T. ANDO AND I. AMEMIYA, Almost everywhere convergence of prediction sequencein L, (1 < p < co), Z. Wahrsch. Verw. Gebiete 4 (1965), 113-120. 2. R. R. BAHADUR, Measurable subspaces and subalgebras, Proc. Amer. Math. Sot. 6 (1955), 565-570. 3. R. E. BARLOW,D. J. BARTHOLOMEW, J. M. BREMNER,AND H. D. BRUNK, “Statistical Inference Under Order Restrictions,” Wiley, New York, 1972. 4. H. BAUER, “Wahrscheinlichkeitstbeorie und Grundziige der MaBtheorie,” de Gruyter. Berlin, 1968.
ISOTONIC APPROXIMATION IN L,
223
5. H. D. BRUNK, Best fit to a random variable by a random variable measurable with respect to a u-lattice, Pacific J. Math. 11 (1961), 785-802. 6. H. D. BRUNK,On an extension of the concept conditional expectation, Proc. Amer. Math. Sot. 14 (1963), 298-304. 7. H. D. BRUNK, Conditional expectation given a u-lattice and applications, Ann. Math. . Statist. 36 (1965), 1339-1350. 8. H. D. BRUNK, Uniform inequalities for conditional p-means given u-lattices, Ann. Probability 3 (1975), 1025-1030. 9. H. D. BRUNK AND S. JOHANSEN,A generalized Radon-Nikodym derivative, Pacrpc J. Math. 34 (1970), 585-617. 10. R. G. DOUGLAS,Contractive projections on an L,-space, PaciJic J. Muth. 15 (1965), 443-462. 1I. R. L. DYKSTRA, A characterization of a conditional expectation with respect to a ulattice, Ann. Math. Statist. 41 (1970), 698-701. 12. P. GXNSSLERAND W. STUTE,“Wahrscheinlichkeitstheorie,” Springer-Verlag, New York, 1977. 13. A. N. KOLMOGOROV, Stationary sequencesin Hilbert spaces, Bull. Math. Univ. Moscow 2, No. 6 (1941). [In Russian]. 14. D. LANDERSAND L. ROGGE,Characterization of p-predictors, Proc. Amer. Math. Sot. 76 (1979), 307-309. 15. S. C. MOY, Characterizations of conditional expectation as transformation on function spaces, Pacific /. Math. 4 (1954), 47-63. 16. J. NEVEU, “Discrete Parameter Martingales,” North-Holland, Amsterdam, 1975. 17. J. PFANZAGL, Characterization of conditional expectations, Ann. Math. Statist. 38 (1967), 415421. 18. E. SPARREANDERSENAND B. JESSEN,Some limit theorems on set functions, Math. Fys. Meddr. 25, No. 5 (1948). 19. A. WALD, “Statistical Decision Functions,” Wiley, New York, 1950. 20. N. WIENER, “Extrapolation, Interpolation and Smoothing of Time Series,” Wiley, New York, 1941.
OF APPROXIMATION
THEORY
Isotonic
31, 199-223 (1981)
Approximation
in L,
DIETER LANDERS Mathematisches Institut der Universit6t .zu K&‘n, Weyertal 86-90, D-5000 K&I 41, West Germany
L. ROGGE Universittit-Gesamthochschule Duisburg, Posflach 10629, Fachbereich IlfMathematik, 4100 Duisburg I, West Germany Communicated by P. L. Butzer Received October 4, 1979
1. INTRODUCTION AND NOTATION Let (Q, d, P) be a probability space and 0 < s < co. Denote by L&R, M’, P) the system of all equivalence classes of J-measurable functions X: O-+ IR with IlXll, := [l lXISdP]“” < co. For s > 1 the space L,(k4.&‘,P) endowed with the norm 11IISis a uniformly convex Banach space. Let SP c ~8’ be a u-lattice, i.e., a system of sets which is closed under countable unions and intersections and contains 0 and R. A function X: R + R is P-measurable if {X > a} E 9 for all a E R. Denote by L,(y) the system of all equivalence classes in L&2, J, P) containing an ymeasurable function. Then LS(.P) is a closed convex set. Let X E L,(Q, J, P); an element YE t,(P) is called a conditional s-mean of X given the o-lattice 4p, if /IX - Ylls = min(/IX - Z(I, : Z E L,(P)}. For s > 1 the uniform convexity of L, guarantees existence and uniqueness of a conditional s-mean of X given 9, denoted by PYX. The above minimization problem is very important, since it allows one to treat approximation problems under order restrictions. For instance, let P be a probability measure on the Borel-field B” of the n-dimensional Euclidean space R”. Let J be the family of all functions YE L&T?“, IB”, P) which are monotone nondecreasing in each component. Define 4p := {C E [B” : x E C, x ,< y 2 y E C}, where x < y means x, < yi for all components. Then 9 is a 199 0021-9045/81/030199-25502.00/O Copyright 0 1981 by Academic Press, Inc. All rights of reproduction in my form reserved.
200
LANDERSANDROGGE
a-lattice and A is the system of all g-measurable YE L,(lR”, iB”, P). Hence for each XE L&R”, iB”, P) the conditional s-mean of X given 9 is the unique function which is monotone nondecreasing in each component such that the ]] (I,-distance from X is minimal among all functions which are monotone non-decreasing in each component. For s = 2, approximation problems of this type are highly relevant to the theory of isotonic regression. Illuminating examples and the broad theory of isotonic regression can be found in the book of Barlow et al. [3]. Although much work has been done for the case s = 2-the statistical inference under order restrictions-little is known for the case s # 2. The concept of conditional s-means given a a-lattice was introduced in Brunk [8]. For s = 2 it is the usual conditional expectation given a u-lattice, defined on L, (see [5]). If L? is a u-field one obtains the s-predictors in the sense of Ando and Amemiya (see [ 1I), which coincide for s = 2 with the conditional expectations. The theory developed here traces back to Kolmogoroff [ 131 and Wiener [20], it is intimately related to Wald’s decision theory and plays an important role in Non-linear Prediction, Filtering, Regression and Bayes Estimation. In a forthcoming paper we apply this theory to define and estimate a “natural” median (i.e., a “reasonable” selection of a median). At first we collect some properties of c on L, and prove an integral inequality for the conditional s-means (Theorem 2.11.) which is essential for the further considerations (see Section 2). In Section 3 we extend the operator c from L, to L,-, as the unique operator preserving the property of monotone continuity. An example shows that even for L/ = (0, Q} a monotone continuous extension to larger spaces L, (r < s - 1) is not possible in nearly all cases. In Section 4 we show that FX convergesP-a.e. to PymX if X E L,-, and the a-lattices 9” increase or decreaseto the a-lattice L& . This result contains and extends other results in this direction. For s = 2 and u-fields it is a classical martingale theorem and yields a result of Brunk and Johansen (91, for s # 2 and a-fields it extends a result of Ando and Amemiya [ 11. In Section 5 we prove maximal inequalities, i.e., inequalities for P{su~,,~ leXl> a] and I su~,,~ ]P,“XI’ dP. Our inequalities applied to s = 2 and u-fields yield the classical maximal inequalities; applied to s # 2 and u-fields they lead to much sharper maximal inequalities than those of Ando and Amemiya (see [ 11). For u-lattices they seemto be the first results in this direction. In Section 6 we start with a characterization result for conditional s-means with respect to u-fields (Theorem 6.1). This result, applied to s = 2, yields the characterization results for classical conditional expectations of Bahadur [2], Douglas [lo], Moy [15] and Pfanzagl [ 171. Then we characterize
201
ISOTONIC APPROXIMATION IN L,
conditional s-meanswith respect to a-lattices (Theorem 6.4). For s # 2 this is the first characterization result of conditional s-means given u-lattices. For s = 2 we prove a further characterization result (Theorem 6.3) which contains Dykstra’s [ 1l] characterization for conditional expectations given a u-lattice. 2. PROPERTIES OF Pf’I L,
Let (0, ,oP,P) be a probability space. During this section let 9 c & be a fixed u-lattice and 1 < s < co. The conditional s-mean T = c: L, -+ L, has, according to Brunk [8], the following properties T is idempotent; i.e., T( TX) = TX,
(2.0)
T is positive homogeneous;i.e., T(uX) = aTX,
a >O;
T is translation invariant; i.e., T(X + b) = TX + 6,
(2.1)
b E R,
(2.2)
and fulfills
!-(X-TX)s-ITXdP=O; I (X-TwZdP
(2.3)
if
Z EL,(Y),
(2.4)
where aS-’ = l~(~-‘sign a for a E IR. Relation (2.4) aplied to Z E +l and Z E -1 implies that T is s-expectation invariant; i.e., (X - TX)s-‘dP I
Using that XLTX
> (X - TXF
= 0.
(2.5)
TX with > if TX # 0, we obtain from
(2.3) T is s-strictly monotonic at 0; i.e., xs-L’TXdP>
0
for
(2.6)
TX# 0.
As T = c is a nearest point projection onto a closed convex set of the uniformly convex space L, it is well known that T is continuous in the norm topology of L,.
(2.7)
202
LANDERS ANDROGGE
Brunk [8] has shown that a
(2.8)
Let X < Y and define Z, = TX A TY, Z, = TX V TY. Applying Lemma 7.2(iii) pointwise to a = Y, b = TY, c = X, d = TX we obtain by integration
Since Z,, Z, E L,(9) we obtain ]]Y - TYI(, = /] Y - Z,ll, Z, = TY, i.e., TX < TY. From (2.7) and (2.8) we obtain that T is monotone continuous; i.e.,
and hence
(2.9)
X, r X (X, 1 X) implies TX, T TX (TX, 1 TX). Let 2 := {c : C E 9} be the dual u-lattice of 9. Since L,(9) = --L,(g) follows directly from the definition of s-means that PTX = -PY(-X),
XE L,.
it
(2.10)
Now we prove an integration inequality which is an important tool in the following sections. For s = 2 this was proven in [3, p. 3421 by other techniques. Denote by [B the Borel-field of I?. THEOREM
Y-measurable.
2.11. Let rp: I? -+ [0, 00) be B-measurable and Z: R -+ R be Then we have for all X E L,
if the integral exists. ProoJ W.1.o.g. we may assume that Z is bounded. By the usual techniques it can be seen that it suffices to prove the assertion for nonnegative bounded functions p with bounded derivative. Let such a q be given and it4 := suplen l@(t)l < co.
(i) We prove the assertion for Z = 1, with C E 9: Let Y,:=~X+(a/M)y,oP~X,OOaa l.Then Y, = ly, 0 PYX,
where y,(t) := t + (a/M) p(t), t E I?, 0 < a < 1.
(1)
ISOTONIC
APPROXIMATION
203
IN L,
Since w,: R -+ R is monotone increasing, we obtain from (1) that Y, is y-measurable for all 0 < a < 1.
(2)
Define Z, :=PyX+
(a/kf)l,q70
PYX,
O
1.
Since q~> 0 we obtain with (2) for all b E R, 0 < a ( 1 {Z,>b}={I$%>b}U(Cn{Y,>b})W and hence Z, is y-measurable for all 0 < a ( 1.
(3)
The function da) := IM- Z&9
O,
attains its minimum at a = 0, because Z, = P,“X and Z, EL,(Y) a E [0, 1). Hence O
for all
[(X-P~X)~l,a,oP~XdP,
which implies the assertion for Z = 1,. (ii) The assertion holds for all non-negative bounded Y-measurable functions by (i), since every non-negative Y-measurable function is the monotone limit of non-negative y-measurable simple functions. (iii) Now let Z be p-measurable and bounded. Since Z = Zt - Z-, it suffices, according to (ii), to prove that -
i
(X - P:XFZ-q
0 PYX dP & 0.
(4)
The function I&) := rp(--t), t E R, is a non-negative bounded B-measurable function. Hence (4) is fulfilled if we show i
(-X - (-cX)FZ-
I//(-P,“x> dP & 0.
(5)
Let 2 = {C: C E 9} be the dual lattice of 5R. Since Z- is g-measurable and -PyX = PT(-X) by Property 2.10, (ii) applied to the u-lattice @ yields (5).
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3. EXTENSION AND PROPERTIES OF FILL,-,
Up to now the operator Py is defined on the domain L,. For s = 2 it is known that Pi;” can be extended from L, to L, . It is our aim to extend for general s > 1 the conditional s-mean Py as a monotone continuous operator from L, to larger spacesL,, more exactly to choose r as small as possible. For instance, can we always extend the conditional s-mean from L, as a monotone continuous operator mapping L, into L 1? It turns out that we can extend Py from L, to L,- I but, even in extremely simple cases, not beyond L,- , . This shows, for instance, that for s > 2 we cannot extend to L, but for 1 < s < 2 we can extend even beyond L, . We write Pf’/L in order to indicate that Py is defined on L c L, ~, . DEFINITION 3.1. (i) Let f!-i be the system of all XE LS-,(.R, AZ?‘, P), which are bounded from below by a function of L,. For X E f!- , define
*x
:= ljz P,“(XA n).
We remark that this limit exists since Py/L, is monotone and X A IZ, n E N, is a nondecreasing sequence in L,. It is easy to see (use Property (2.9)) that P,“lZ!,- , is a monotone extension of PYL, . (ii) For X E L,-l define Pf!X := ljz P,“(X V (-n)). We remark that this limit exists since Pfjf?+, is monotone and XV (-n), n E N, is a non-increasing sequencein f?-, . Since it turns out that P,“/L!,-, is monotone continuous (this is proven in Theorem 3.2), PSy’L,- 1 is an extension of c/E, _, and hence of PF/L, . THEOREM 3.2. Let g c A’ be a o-lattice and 1 < s ( co. Then the operator PT/L, _ 1 has thefollowing properties:
(PO) (PI) (P2) (P3) (P4) (P5) (P6)
e maps L,-, into LSpI, pf”IL,-, is idempotent, c/L,-, is montone, c/L, _, is positive homogeneous, c/L,-, is translation invariant, c/L,- 1 is monotone continuous, c/L,-, is s-expectation invariant,
ISOTONIC
APPROXIMATION
205
IN L,
(P7) J(x-~*z~o~XdP
is y-
(P8) I(X-eX)s-IpopYXdP=O measurable and the integral exists.
is B-
if XEL,-,,
(0: R-+R
Proof. We prove the properties above for Py/l!- r, using the corresponding properties of Py/L,. Using the then proven properties of Pf’/Q!,- r, one obtains in a similar way the stated properties of Py/L,-, .
(PO) Let X E I?-, . Then X, := X A IZE L,. As PF/L, is s-expectation invariant (see Property 2.5) we obtain P[(X,-PyXJq
=o
(1)
AS(X,-P~X,~~(X-P~X,)S-‘~~~XE~,_,,P~X,EL,,(~)~~~~~~~ o
Since (X - PyX,p
nEN
< co,
1 (X - PYXF,
(2)
(2) implies
o
< 00
and hence X - PTX E L,-, . As L,-, is a linear space and X E L,-, , consequently PYX E L,- , . Since X> YE L, implies P,“X> PYY, we get PTXE Q!,-,.
Property (Pl) follows directly
from (2.0) and the definition of
Pyp-,.
Property (P2) follows from (2.8) and the definition of PF/lI?S-, . Property (P7) W.1.o.g.we may assumethat Z is bounded. By the usual techniques it can be seen that if suffices to prove the assertion for nonnegative, continuous and bounded functions (o. Let now X E !i!!,-, and X, := X A n. Then X, E L, and therefore Theorem 2.11 implies
I (X, - P:X,pZrp
0 P,“X, dP < 0,
n E N.
As q is continuous, v, o PYX,, -+ p o P,“X by definition of P,“X. PYX E L,- 1 according to (PO), it is easy to see that
Since
is bounded by an integrable function. Hence the Theorem of Lebesgue implies the assertion.
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Property (P6) follows directly from (P7). Property (P5) W.1.o.g. we prove only the decreasing case. Let X,, E L,-, with X, 1 XE 2,-r. Then by (P6)
((xn-PyxnpdP=o,
nEN.
(3)
By (P2) we obtain PYX < PTX, ( CX, . Hence the Theorem of Lebesgue implies by (3) s--l
x-
As 1 (X - CXpdP PYX = lim, ENCX”.
p& PYXn
dP=O.
(4)
= 0 and PYX < lim,,, PYX,, , (4) implies that
Properties (P3) and (P4) follow immediately from (P5), using the corresponding properties of P$, (see Section 2). Property (P8) follows from (P7) applied to q’+ and (o- with Z E 1 and Zr-1. The preceding theorem shows that there exists a monotone continuous extension of Py/L, which maps L,- i into L,- r . The following remark shows that, even for 9 = (0, a}, such an extension is not possible for any larger space L, (r < s - 1). Remark 3.3. Let (0, d, P) be a probability space and 1 < s < co. Assume that L,#L,-, for some O
Let X,, :=X A n E L,. Then X,, T X. If there exists a monotone continuous extension from c/L, to L,, then c, = P,“x T PFX = c. Assume that c < co. Since c/L, is s-expectation invariant we have 0=1(X,,-c,FdP>j+(X,-cFdP>-co.
Since X, T X, this implies 0 > J”(X - c)” dP > -a~. Hence X E L,- , , contradicting our assumption. For each X E L,-, we give now a characterization of the conditional smean eX as the unique g-measurable function fulfilling two integralinequalities. For the case s = 2 this is known for X E L, (see [3]); it seems to be unknown even for s = 2 if X E L, .
ISOTONICAPPROXIMATION IN L,
207
THEOREM3.4. Let X E L,- 1 and cpbe a strictly increasing function such that (Do PYX is bounded. Then Y = P,“X if and only if YE L,- ,(9), rp 0 Y is bounded and (i) (ii)
1(X--YF Z dP < 0 for all bounded 4P-measurable Z; 1(X-- YFqo YdP=O.
Proof. The direction “only if’ follows from (P7) and (P8) of Theorem 3.2. For the converse direction it suffices to prove that if Y,, Y, E L,_ ,(9) are two functions fulfilling (i) and (ii), then Y, = Y,. By (ii) we have
I (x-
Yj)+p
o Y,dP=O,
j= 1,2.
Since q o Yj, j = 1, 2 are bounded y-measurable functions we obtain, together with (i), f ((YI - XF
+ (X - &j+(v)
0 Y, - rp 0 Y2) dP < 0.
Since rp is increasing, it is easy to see that ((Yl - xp
+ (X - Y*)s--I)((o 0 Y, - (00 Y*) > 0.
Hence rpo Y, = rpo Y, P-a.e., whence Y, = Y2 P-a.e. COROLLARY3.5. Let YE L,-,(y)
(i) (4
X E L,-, .
Then
Y = PFX
if
and
only
if
and I,(X-Y)“-‘ldP
YFdP=Ofor
all aE R.
Proof. The direction “only if’ follows from (P7) and (P8) of Theorem 3.2. For the converse direction we have to prove (i) and (ii) of Theorem 3.4. Since v(B) := J (X- wl, 0 Y dP, B E [B, is a signed measure which is zero on the system {[a, co), a E IR} according to (ii), it is zero on IB also. This implies (ii) of Theorem 3.4 even for all measurable functions cp,for which the integral in (ii) exists. Since every non-negative bounded 9measurable function is the monotone limit of non-negative Y-measurable simple functions, (i) implies, that condition (i) of Theorem 3.4 holds for all non-negative, bounded, y-measurable functions Z. Let Z be a bounded 9-
208
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AND ROGGE
measurable function. Then Z + a is a non-negative bounded p-measurable function for suitable a and hence
i
(X- Y,P(Z+a)dP~O.
Consequently we obtain (i) of Theorem 3.4. Now we give some further properties of the extended Py. Since we obtain from Property (P7) of Theorem 3.2 with L,(-q = -L,(@, Z=-1, that for XEL,-, (P9) j”,,(X-cXpdP>O if DEg, CE(PyX))-‘IB, where I3 denotes the Borel-field of R. Using the monotone continuity of Py/L,+ 1, Property 2.10 implies (PIO) PYX= -PY(-X), XE L,-,. Since X E L,- I and Y = PYX fulfill (i) and (ii) of Corollary 3.5, we obtain for each u < 1 that X0 =X - uP:X and the y-measurable function Y, = (1 - a) eX fulfill (i) and (ii), too. Hence Corollary 3.5 implies: (Pll) Py(X-uP:X)=(l-u)PyX, a,< l,XEL,_,. Since Z$? is monotone we have P,“(--1x1) ,< PyX< P,“lXl. Hence (PlO) implies (P12) IT4 < max(C(Xl, FIXI), XE L,-, . We remark that for o-lattices it is not true that lcX1 (P,“lXl; this can be seen by simple examples even in the case s = 2. Using (P2) and (P8) it is easy to see that for each interval Zc IR (P 13) X E Z P-a.e. implies P,“X E I P-a.e. Now we prove a convexity inequality for conditional s-means. (P 14) Let Z c R be an interval and q: Z + R be a non-decreasing continuous and convex function. If X, q 0 XE L,-, and XE Z P-a.e. then a, 0 PfcX < PY($o 0 X). ProoJ
Since ~1is continuous and convex there exist a,, b, E R, n E Id,
such that
P(X) = “,t~ (w + 4)
for all
x E I.
Since v, is non-decreasing, w.1.o.g.a, > 0. From (P2), (P3), (P4) and (P13) we obtain P,“(u, o x) > sup u,P:X nefh
+ b, = q 0 PYX.
Q.E.D.
ISOTONIC APPROXIMATION IN L,
209
Finally we prove for each r > s - 1 (P15) XEL,=+-PYXEL,. According to (P12), we may w.1.o.g. assume that X > 0. Since Xr’(‘-‘) E L,-, , (PO) implies Py(XV”-“) E L,-, . As, by (P14) (PYX)rlcs-l) < P$‘(X”‘“~“) E L,- i, we obtain PYX E L,. s 4. CONVERGENCE A.E.OF PFX In this section we give a general a.e. convergence theorem for conditional s-means P,‘;lX for all X E L,- i . For the case s = 2, it is a result of [9]. For the special case of o-fields and for X E L, this was proven in [l]. The techniques used here, are related to the methods of Sparre et al. [ 181 and Brunk and Johansen [9]. If Yn c d, n E N U {co }, are a-lattices, Yn, n E iN, increases [decreases] to pa if 9” c ik7,+,[pn 2 49,+i] and pa is the o-lattice generated by
unsN zwm= nnd3
THEOREM 4.1. Let (Q, z.4, P) be a probability space and 1 < s < 00. Let Yn c &‘, n E IN, be o-lattices increasing or decreasing to the o-lattice pm. Then
P-a.e.,
PFX + P,“,X, forallXEL,-,(&sd,P).
Proof. We prove only the decreasing case, which is somewhat more complicated. The proof for the increasing case runs similarly. LetX,=~X,nE~:=NU{co}.Let~~:={C:CE~,}.Accordingto properties (P7) and (P9) of Section 3, we have (X-apdP
if
aElI?,C,EY~,nER,
(1)
and
Define X= bneN X, and x= hm,,, X,; then & X are IP,-measurable. Let C E Ym and a E IF?be fixed and choose a sequence E, 1 0. Define for mQn
210
LANDERS
AND ROGGE
According to (I), we have
Ic!r (X-(a+c,)FdP
(3)
Since CL=, Cr,, = en {infmGiGrXi ( a + em}=: Cr and CT t,+, C” := C {inf,>, Xi < a + cm} we obtain, from (3),
n
J (X -
(a + E,))~
m E N.
dP ,< 0,
(4)
Cm
Since lc,+
msN
1Cn{g(at'
(4) implies, using the Theorem of Lebesgue
Jix
if
uE IR,CEYa.
(5)
if
bE I?,DEgm.
(6)
In the same way we obtain (X-bFdP>O, J(W>6jfTD
Now we show, using the relations (l), (2), (5), (6) that RX,,
X,
P-a.e.
This directly implies the assertion. We prove only X,
has P-measure zero, Relation (2) applied to n = co and D, = {X < a} E .@ff yields
J(X-bpdP>O.
(7)
E
Relation (5), applied to C = {X, > b} E ip, yields
JE (X--aFdP
> (X - b)“L on E, (7) and (8) imply P(E) = 0.
04)
211
ISOTONIC APPROXIMATION IN L,
5. MAXIMALINEQUALITIES
AND NORMCONVERGENCE
OF P:X
In this section we prove two maximal inequalities and give an application to norm convergence. THEOREM 5.1. Let (SJ,&, P) be a probability space and 1 < s < 00. Let 9” c ,?p, n E N, be an increasing or decreasing sequence of o-lattices. For non-negative X E L,- , we have
ProoJ:
Let X,, = FX,
n E N. It s&ices to prove for each n E N that
As for decreasing sequences9,) n E N, the finite sequencepn, Yn-, ,..., 9, is increasing; it suffices to prove (1) for the increasing case. At first we show that
1
(X-aFdP>O,
a > 0.
(2)
ImaXl
Let Ai = {X,
(X-apdP>j A,
(X-X,)“L’dPaO. Ai
As CrrIA, = (max , GiGnXi > a], this implies (2). Applying Lemma 7.2(i) and (ii) pointwise to x =X(w) for all o E {maxICiGn Xi > a}, we obtain by integration, using (2), inequality (1) COROLLARY 5.2. Let (l2, s’, P) be a probability space, 1 < s < 00 and r > s - 1. Let Yn c J/, n E IN, be an increasing or decreasing sequence of alattices. Then we have for XE L, that
where u,,, = (2”-*I [r/(r - s + l)])“‘“-” X is non-negative and 6 = 2 elsewhere.
and 6 = 1 ifall Yn are o-fields or if
212
LANDERS AND ROGGE
ProoJ First we prove the assertion for X> 0. We use the following well known lemma [ 12, p. 23 11: If <, q: R -+ [0, co] are d-measurable with
P{r > s} < (l/a)j
CdP
lV>E)
for all
E > 0,
(“1
then P[ya] < [a/(a - l)]” P[ca] for all a > 1. Let q := (supnENPpX)‘-‘, r= 2ls-21~s-’ ) &=a$-’ and a = r/(s - 1) > 1. Then (*) is fulfilled according to Theorem 5.1. Hence the lemma cited above implies the assertion for X> 0. Hence the assertion is true for o-fields and arbitrary XE L,, using ]eX]
We remark that for the case s = 2 and a-fields we obtain in Theorem 5.1 and Corollary 5.2 exactly the classical maximal inequalities with the same bounding constants. For a-fields 9n and X E L, Ando and Amemiya [l] proved that P
[
“,llR lPs”“Xl”
1
The constants a, given by Ando and Amemiya are much larger than our constants u,,, specialized to the case r = s. For s = 2 we have u,,, = 4 and a, = 208 and for s > 2 there holds (2szQ-
l < u,.
Observe that 1 < u,,, = (21s-2’s)s’(s-‘) + co with s+ co. For u-lattices the preceding results of this section seemto be the first in this direction. COROLLARY 5.3. Let (f2, &, P) be a probability space, 1 < s ( co and r > s - 1. Let Yn c H’, n E N, be a sequence of a-lattices increasing or decreasing to the a-lattice ip, . Then for each X E L, we have
Proof. According to Theorem 4.1 and Corollary 5.2 the assertion follows for r > s - 1 by the Theorem of Lebesgue. For r = s - 1 it suffices, according to Theorem 4.1, to prove that JPSyXjSml, n E N, is uniformly integrable. Since IP?XJS-’ < (PF IXJ)s-’ + (PF (XI)‘-’ and the sum of two uniformly integrable sequences is uniformly integrable (use e.g. [4,
213
ISOTONIC APPROXIMATION IN J!,,
Korollar 20.3, p. 901) we may w.1.o.g. assume that X > 0. Let X, = P%X, n E N. We obtain from Lemma 7.2(i) and (ii), applied pointwise for w E {X, > a} to a =X,(U), x = X(o)-since J,X,>aj (X- X,pdP = 0 according to (PS) of Section 3-that J’lX,>al
xi-’
dP < 21s-*’
I lX,>al
Xs-’ dP,
a > 0.
(+>
Since ~up,,~ P{X, > a} +a+m 0 by Theorem 4.1, we consequently obtain from ( + ) that Xi-‘, n E N, is uniformly integrable. 6. CHARACTERIZATIONS OF Pf’ In this section we characterize the operators Py for u-fields and o-lattices. We start with a characterization result for o-fields and arbitrary s > 1. For s = 2 this leads to a characterization for classical conditional expectation operators which is a common generalization of the results of Bahadur [2], Douglas [lo], Moy [ 151 and Pfanzagl [ 171. THEOREM 6.1. Let 1 < s ( CC and s - 1 < r < co. Let T: L,(Q, J, P) --+ L,(Q, &, P) be an operator which is
(i) (ii) (iii)
homogeneous,
(iv> (v)
monotone,
translation
invariant,
idempotent, s-expectation invariant.
Then there exists a sub-a-field XE L,.
3 c ~2 such that TX =PTX
for all
Proof. Let ,5?:= {A E JY: Tl, = lA}. Since T is homogeneous and translation invariant, A E J implies Tl,= 1 - Tl, = 1 - 1, = 1,. Hence 9 is a a-field according to Lemma 7.3(ii). Since T, e: L, -+ L, are monotone continuous operators according to Lemma 7.3(i) and Property (P5) of Theorem 3.2 it suffices to show that TX= P,“X
for bounded X.
We remark that according to (i), (ii) and (iv) TX is bounded if X is bounded. Applying (iv) of Lemma 7.3 to Z and -Z we obtain
I (X640/31/3-2
TXpZdP=O
for
Z E L,(3).
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LANDERS AND ROGGE
Hence we can apply Theorem 3.4 with Y = TX and the strictly increasing function q(t) = t, t E R. This yields TX = PFX for bounded X. Easy examples show that none of the five conditions in Theorem 6.1 can be omitted without compensation. Another characterization result for conditional s-means given a a-field can be found in [ 141. In the case s = 2 s-expectation invariance of an operator T means s TX dP = s X dP. Hence for s = 2 monotony and s-expectation-invariance of T trivially imply that T is weak 1)J/,-reducing; i.e., IITX-
TYII, G 11X- YII,
if
X< Y.
Furthermore each linear operator with I]TX/I, < ]lX]l, is trivially weak 1)/Iireducing. Therefore the following corollary contains the characterization results for conditional expectation operators given in [2], [lo], [ 151 and. (171. COROLLARY 6.2. Let 1 < r < co and T: L,(R, ~4, P)-+ L,(sZ, ~4, P) be an operator which is
(i) (ii) (iii) (iv)
homogeneous, translation
invariant,
idempotent, weak 1111,-reducing.
Then there exists a sub-o-field XE L,.
9 c J
such that TX = PfX
for
ail
Proof. According to Lemma 7.4 T is monotone and 2-expectation invariant. Hence Theorem 6.1, applied to s = 2, implies the assertion. The only characterization result, known to the authors, concerning ulattices is the result of Dykstra [ 1l] for the case s = 2. He shows that an operator T: L&Q, d, P) -+ L&2, @‘, P) which is
(i) (ii) (iii)
positive homogeneous, ]I I(,-reducing; i.e., IITX - WI2 < IIX - Yll,, idempotent, (iv> monotone, (v) Z-expectation invariant; i.e., ] TX dP = j X dP, (vi) 2-strictly monotonic at 0; i.e., ] XTX dP > 0 if TX # 0
is the conditional expectation operator c with respect to a suitable u-lattice LFCC. We remark that Dykstra requires instead of (vi) a slightly stronger condition, but he uses in his proof only condition (vi) [3, p. 3221. Dykstra
ISOTONIC
APPROXIMATION
215
IN L,
furthermore gives an example showing that conditions (i)-(v) above are not sufficient for his characterization result [3, p. 3261. In the characterization results for conditional expectation operators given a a-field, only one integration condition besides algebraic conditions is used. This is an advantage since the authors believe that algebraic conditions are nicer and easier to verify than integration conditions. Dykstra’s characterization result for a conditional expectation operator, given a u-lattice, uses three integration conditions. The “strongest” of his integration conditions is the property (ii), namely, 11II,-reducing. In the following theorem we show that 11I/,-reducing can be replaced by the algebraic condition translationinvariance. Furthermore monotony and 2-expectation invariance is weakened to weak /I I/,-reducing. Since 11iI,-reducing and II II,-expectation invariance imply translation invariance (see [3], Proposition 7.2, 2”, p. 324) the theorem below contains the result of Dykstra. THEOREM 6.3. Let 1
T: L,(G!, &, P)+ L,(R, zZ,P)+
(i) positive homogeneous, (ii) translation invariant, (iii) idempotent, (iv) weak I/ /I ,-reducing, (v)
2-strictly monotonic at 0 for XE L,.
Then there exists a o-lattice 9 c J/ such that TX = PyXfor
all X E L,.
Proof. Since according to (iv) T is II iI,-continuous for monotone sequences, it suffices to prove TX = PYX for X E L,. According to Lemma 7.4 the operator is 2-expectation invariant and monotone and hence according to Lemma 7.3 there exists a u-lattice g c & such that
c
(X-TX)ZdP
for X E L,(d),
Z E L,(Y).
According to Theorem 3.4, applied to s = 2, X:=X
(1)
- TX, Y = 0 and
q(t) = t, (1) implies that Pf(X - T(X)) = 0.
(2)
T(X - PFX) = 0
(3)
Furthermore we have
because T(X - KX) E L,(Y) implies l (X - PTX) T(X - Pf”X) dP < 0 and (v) implies I (X - CX) T(X - CX) dP > 0 if T(X - KX) # 0.
216
LANDERSANDROGGE
Now we shall show that
IITXIII= IIe%
XEL,.
(4)
According to Lemma 7.4, the operator T is ]] I),-reducing. Using (2), (3) and the fact that e is ]] III-reducing, too, we obtain:
Il~~lI~=Il~-~~-~~~ll,~ll~~-~~~-~x)lI,=ll~~Il, and
IITXII, =11X-(X-
T’)(l,hll~X-P?(X-
Wll, =llP~Xll,.
Hence (4) follows. Now (4) and translation invariance of T and PT imply ~~--u[~,=l~TX--n~~, for XEL,, aElF?; i.e., ~I~-ulPl(dx)= x - a ] P,(dx) for a E F?, where P, is the distribution of P, X and P, the distribution of TX. Hence according to Lemma 7.5, P, = P, and therefore especially !’ (TX)’ dP = ( (PyX)‘dP.
(5)
Now we obtain by (5) and (I), that 11TX - EXll;
= 2 [ (PFX)’ dP - 2 j” TXPYX dp = 2 =2
XP’;yX dP -
I
i
TXPyXdP
1
(X-TX)PFXdP
and hence TX = PYX. Dykstra’s example cited in [3, p. 3261 (52= { 1,2}, P{ 1) = P(2) = 4, if X(1) > X(2)), TX = X if X(1)
(i) (ii) (iii)
positive homogeneous, translation invariant, idempotent,
ISOTONIC
APPROXIMATION
217
IN L,
(iv)
monotone, (v) s-expectation invariant, (vi) weak s-monotonic at 0; i.e., I Xs-=--‘TXdP > 0 for X E L, TX > 0 and fulfills: (vii) T(X - f TX) = 4TX.
with
Then there exists a a-lattice 9 c JCZsuch that TX = PTXfor all X E L,. Proof. According to Lemma 7.3(ii), (iii) we have that Y := (A E J/: Tl, = 1, } is a u-lattice and TX E L,(Y) as T is idempotent. Since T, Py: L, + L, are monotone continuous operators, it suffices to show that TX = PrX
for bounded X.
For this, it suffices to prove condition (i) and (ii) of Theorem 3.4 applied to the bounded function Y = TX. Condition (i) of Theorem 3.4 is fulfilled according to Lemma 7.3(iv). Since TX is bounded and y-measurable, Condition (ii) of Theorem 3.4 is fulfilled with q(t) = t if we show that i
(X- TX)“TxTXdP>
0.
(*I
Define inductively y, = 1 and yn+r = (1 + y,)/2. We prove by induction T(X-y,TX)=
(1 -r,>TX.
c**>
For n = 1 this is our Assumption (vii). Assume now that (**) holds for n. Then T(X - y,,+ 1TX) = T(X - y,,TX - )T(X - ynTX)) =~T(x-~,TX)=~(~-QTX=(~-~~+,)TX. Hence (**) holds for (n + 1). Let X2 0, then (1 - y,) TX > 0. Hence, according to (vi) and (**), O<
I
(X-y,TXpT(X-yy,TX)dP
i.e., i
(X- y,TXpTXdP>O.
T(X- y,TX) =
218
LANDERS AND ROGGE
As yn--t 1, we obtain (*) for bounded X > 0. As T is translation invariant we obtain (*) for all bounded X. We remark that conditions (i)-(vi) are even in the case s = 2 not sufficient to characterize s-means. This can be seen from the just cited example of Dykstra.
7. AUXILIARY
LEMMATA
At first we prove a lemma characterizing a system of functions as a system of y-measurable functions where 9 is a suitable u-lattice. This lemma plays on analogous role for a-lattices as Lemma 3 in [ 171for u-fields. 7.1. Let 0 ( r < co and 0 # F c L,(f2, -xf, P). Assume that F the following conditions:
LEMMA
jiiljZls
(i) (ii) (iii)
aX+,BEFfirXEF,a>O,/3ElR,
XA Y,XV
YEFforX,YEF,
X,EF,XEL,andX,,fXorX,1XimplyXEF.
Let 9 := {A E ~4: 1, E F}. Then Ip is a o-lattice and F = L,.(p). Proof.
Since lc,n c. = l,, A lcZ, l,,,cl = l,, V lc, and 1n;=,ckl lnkm,,ck~
1u;=,c, T lu:,w
(ii), (iii) directly imply that 9 is a a-lattice. Now we show F c L,(p). X E F be given then for each /I E R
Let
Y,,:= [4X-P) V 01A 11T lfw:xcwj>41. As (i), (ii) imply Y, E F we obtain from (iii) that {o: X(o) > p} E 9; i.e., x E L,(g).
Finally we have to show L,(9) c F. Let X E L,(P). By (i) and (iii) we may w.1.o.g. assume that X > 0. Since X E L,(Y) we have {cu:X(w) > a} E P and hence l,o:XCw)>a,E F. By (i) and (ii) this implies X, := sup $.
l{w:xCw)>+}: v = 0, l,..., n2”
I
As X,, T X, hence X E F by (iii).
I
E F.
ISOTONIC
APPROXIMATION
219
IN J?.,
LEMMA 7.2. Let 1 < s < 00. Then there hold the following between real numbers:
(i) (ii) (iii) Proof
as-’ +2s-2(x-a~<22s-2xs_=1 as-’ +(x-ap<22-Sx”-’
inequalities
ifa>O,s>2, zfa,x>O,
1
la-bVd\“+(c-bAdl”
Dividing (i) by as-’ it suffices to prove
f(z) := 1 + 2s-72 - l>sL’ < 2s-3s-1 = g(z),
z E R.
(*)
Since f(l)< g(1) and?(z)< g’(z) f or z > 1, (*) is true for z > 1, s > 2. This directly implies that (*) is also true for z < 0. If 0 < z < 1 use differential calculus to show that h(z) = g(z) -f(z) attains its minimum at z0 = { and h(i) = 0. Inequality (ii) follows in the same manner. Inequality (iii) follows using similar techniques. LEMMA 1.3. Let l
(i) (ii) (iii) (iv)
T is monotone continuous, 9 = {A E JS’: Tl, = lA} is a u-lattice, L,(9)
= {X E L,: TX = X},
J(X-TX~ZdP
Proof (i) We show only that X, T X implies TX, f TX if X,, X E L,. The decreasing case runs similarly. Since T is monotone the pointwise limit Y := lim,,, TX, exists and fulfills Y < TX. As T is s-expectation invariant we obtain
!’(X, -
TX,, p
dP = 0
and hence by the Theorem of Lebesgue
c
(X-
wdP=O.
Together with Y < TX and ( (X - TXpdP Y= TX.
= 0 this implies
220
LANDERS AND ROGGE
For (ii), (iii), let F := {XE L,: TX = X}. Since T is positive homogeneous and translation invariant, condition (i) of Lemma 7.1 is fulfilled. To prove condition (ii) of Lemma 7.1 let X, YE F be given. We show XV YE F, the proof for X A YE F runs similarly. Since T is monotone we have T(XV Y)> TXV TY=XV
Y.
As T is s-expectation invariant we have (XV Y-T(XV
0=
Y)FdP,
whence T(X V Y) =X V Y; i.e., X V YE F. Condition (iii) of Lemma 7.1 follows from the monotone continuity of T. Hence Lemma 7.1 implies (ii), (iii). Now we prove (iv) if 0
I I I
(Xl, - T(Xl,))s-r
dP
(Xl, - (TX) lc)ti
dP
1,(X - TX)““L dP.
As T is positive homogeneous and translation invariant we obtain (iv) for all bounded X and Z = l,, C E 9. As T is continuous on monotone sequences we obtain (iv) for all X E Lr(&) and Z = 1, with C E 9. Since every nonnegative y-measurable function is monotone limit of y-measurable simple functions and since T is s-expectation invariant we obtain (iv). LEMMA 7.4. Let 1 < r < co and T: L,(Q, ..M’,P)+ L,(f2, M’, P) be a weak 11[[,-reducing, translation invariant operator with T(0) = 0. Then T is
(9
(ii) (iii)
monotone, 2-expectation invariant, /I ([,-reducing; i.e., ([TX-
TYIJ, < IIX-
YJI,.
ISOTONIC
APPROXIMATION
IN L,
221
Proof. At first we show that T is 2-expectation invariant. Since 7Q = 0 we obtain that
Now let X E L,(jQ) with 0
c~~~ (2)
Hence the two inequalities in (2) are equalities. The first of them implies c = I TX1 + Ic - TX1 and hence 0 < TX < c, therefore the second implies 1‘TXdP=IITXII,=IIXJI,=J-XdP. Since T is translation invariant, 2-expectation invariance holds for all bounded X. Since T is weakly 11III-reducing we obtain that
if X, T X or X, 1 X.
IITX, - T’ll, -+0
(3)
Let X E L,(J) be bounded from below. Then there exist bounded functions X, with X, T X. Since IX,, dP = s TX, dP, we obtain from (3) (jXdP-jTXdP
(=Fg
IjX”dP-jTXdPl
< &$lTX,-TXII,=O.
Hence (X dP = l TX dP for all X E L&f’) which are bounded from below. Since each integrable function is a decreasing limit of integrable functions which are bounded from below, we obtain in a similar manner that s TX dP = j X dP for all X E Lr(~), i.e., (ii). Now we prove that T is monotone. Let X, YE L&t’) with X < Y. Since T is 2-expectation invariant and weak II /,-reducing we obtain j(Y-X)dP=j(TY-TX)dP
222
LANDERSANDROGGE
Hence we have equality, whence TY - TX = 1TY - TX(; i.e., TX < TY. Since T is monotone and weak 11(I,-reducing, T is (1I),-reducing:
LEMMA
1.5.
Let P,, P, be two p-measures on the Borel-field of R. If Ix--alP,(dx)=jIx-alP,(dx)
for all a E R, then P, = P,. Proof It suffices to prove P,(-a~, y] = Pz(--03, y] for all y E R. For all E > 0 we have by assumption
I”x-
YI - lx- (Y + E)l)Pl(dx)= j?lx - YI - Ix--- (Y + c)I)P,(dx);
i.e.,
E[-q--co;Yl + p11.Y+ 6 =)>I+ J (Y.
= E[-P,(-Co,
y] + P,[Y + E, aI1
(Ix-YI-Ix-(Y+~)I)Pl(~) Y+E)
+ J(y,,,+c)(Ix - Yl - Ix - (Y + E)I)P*(dX)*
Divide by E > 0 and let E-+ 0. Then we obtain
-P1(-q u] + P,(Y,a> = -PA-% Yl + P*(Y,03) and hence -2P,(-CO,
y] - 1 = -2P2(-CO, v] - 1;
i.e., our assertion. REFERENCES 1. T. ANDO AND I. AMEMIYA, Almost everywhere convergence of prediction sequencein L, (1 < p < co), Z. Wahrsch. Verw. Gebiete 4 (1965), 113-120. 2. R. R. BAHADUR, Measurable subspaces and subalgebras, Proc. Amer. Math. Sot. 6 (1955), 565-570. 3. R. E. BARLOW,D. J. BARTHOLOMEW, J. M. BREMNER,AND H. D. BRUNK, “Statistical Inference Under Order Restrictions,” Wiley, New York, 1972. 4. H. BAUER, “Wahrscheinlichkeitstbeorie und Grundziige der MaBtheorie,” de Gruyter. Berlin, 1968.
ISOTONIC APPROXIMATION IN L,
223
5. H. D. BRUNK, Best fit to a random variable by a random variable measurable with respect to a u-lattice, Pacific J. Math. 11 (1961), 785-802. 6. H. D. BRUNK,On an extension of the concept conditional expectation, Proc. Amer. Math. Sot. 14 (1963), 298-304. 7. H. D. BRUNK, Conditional expectation given a u-lattice and applications, Ann. Math. . Statist. 36 (1965), 1339-1350. 8. H. D. BRUNK, Uniform inequalities for conditional p-means given u-lattices, Ann. Probability 3 (1975), 1025-1030. 9. H. D. BRUNK AND S. JOHANSEN,A generalized Radon-Nikodym derivative, Pacrpc J. Math. 34 (1970), 585-617. 10. R. G. DOUGLAS,Contractive projections on an L,-space, PaciJic J. Muth. 15 (1965), 443-462. 1I. R. L. DYKSTRA, A characterization of a conditional expectation with respect to a ulattice, Ann. Math. Statist. 41 (1970), 698-701. 12. P. GXNSSLERAND W. STUTE,“Wahrscheinlichkeitstheorie,” Springer-Verlag, New York, 1977. 13. A. N. KOLMOGOROV, Stationary sequencesin Hilbert spaces, Bull. Math. Univ. Moscow 2, No. 6 (1941). [In Russian]. 14. D. LANDERSAND L. ROGGE,Characterization of p-predictors, Proc. Amer. Math. Sot. 76 (1979), 307-309. 15. S. C. MOY, Characterizations of conditional expectation as transformation on function spaces, Pacific /. Math. 4 (1954), 47-63. 16. J. NEVEU, “Discrete Parameter Martingales,” North-Holland, Amsterdam, 1975. 17. J. PFANZAGL, Characterization of conditional expectations, Ann. Math. Statist. 38 (1967), 415421. 18. E. SPARREANDERSENAND B. JESSEN,Some limit theorems on set functions, Math. Fys. Meddr. 25, No. 5 (1948). 19. A. WALD, “Statistical Decision Functions,” Wiley, New York, 1950. 20. N. WIENER, “Extrapolation, Interpolation and Smoothing of Time Series,” Wiley, New York, 1941.