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Best constant in the decoupling inequality for non-negative random variables
Statistics & Probability North-Holland
Letters 9 (1990) 327-329
April 1990
BEST CONSTANT IN THE DECOUPLING RANDOM VARIABLES
Pawel HITCZENKO
INEQUALITY
FOR NON-NEGATIVE
*
Department of Mathematics, Texas A&M Unioersity, College Station, TX 77843, USA Received September 1988 Revised February 1989
Abstract: A simple proof of the following ~~CX4,~34CYkllp~
inequality
is given:
Pal,
where, for n > 1, X, and Y, are Fn-measurable non-negative 9 n-t. Our proof gives the best possible order of constant.
random
variables
with indentical
conditional
distributions,
given
AMS 1980 Subject Classifications: 60E15. Keywords: Non-negative
random
variables,
tangent
sequences,
In the present paper we obtain the best order of constant appearing in the comparison theorem for the L,-norms, p > 1, for sums of tangent sequences of non-negative random variables. Let (0, S-, P) be a probability space with the ascending sequence (Fn) of sub-u-fields of 9. Two (Sn)adapted sequences ( X,) and (Y,) of random variables are tangent if for each n >, 1 the conditional distributions of X, and Y, given pn_i coincide a.s., i.e. if for every real number t, P( X, > r ) .9_ 1) = P( Y, > t ) gn _ 1) a.s. The following conjecture of Kwapieh and Woyczynski (1986) was substantiated by the author in Hitczenko (1988): For every continuous increasing function @ : Iw + --f R!+ such that Q(O) = 0 and @(2t) G a@(t) for some constant (Y and all positive number t, there exists a constant c such that the inequality
holds for all tangent sequences (X,) and (Y,) of non-negative random variables. The proof was * On leave from Academy
of Physical
Education,
inequality.
based on Davis’s decomposition and the ‘good X inequality’ of Burkholder (1973). Our aim here is to give a much simpler proof of the special (but the most interesting) case Q(t) = tP, p 2 1, of the above inequality (for the concave powers, see Zinn, 1985; Hitczenko, 1988, Remark); a proof which gives the best possible growth rate of the constant. Theorem. Let 1 < p < co. Then for all tangent sequences (X,,) and (Y,,) of non-negative random variables the following is true:
where 11X lip denotes the L,-norm of a random variable X. The constant 3p is best possible (up to an absolute constant, not depending on p). We shall need the following lemma, which comes from Hitczenko (1988) and we include the proof for the convenience of the reader: Lemma. If A,, and B,, are 9”-measurable 52 such that
subsets of
Warsaw,
Poland. 0167-7152/90/$3.50
decoupling
0 1990, Elsevier Science Publishers
P(A,I~l-,)=P(B,I~_,)
a.~.,
n>l,
B.V. (North-Holland) 321
Volume
9. Number
STATISTICS
4
AND PROBABILITY
April 1990
LETTERS
then =p c E(S,P-’ i=l
P(UA,) < 2P(UB,). Proof. Denote by A’ the complement of a set A and put C,=D, C,,=B;n ...nBB,c_,, na2. Then
f’(UA,) < f’(W)
+ f’((U4,)
[=I
=piE(Sr-‘-S,T;‘)E Since
n (f-W;)). E
But NU4,)
- S,C;‘)
n (f-J%>) = P((U4)
n (nC,))
G P(U(4
n CR>>
G CP(4
n Cn>
=E
X,+ i X,+
= CE~(CM4)
which completes
E(XkIFA_,)e
f
E(YklFk_l)q
+:+;I+?), k=r+l
= CNCJ%%
13-I)
= CEI(Cn)f’@n
13-1)
= zP(C,,n
2 k=,+l
B,)= P(UBn),
the proof of the lemma.
q
Proof of the theorem. In our proof we shall use some ideas of Garsia (1973). Assume that p > 1 (since otherwise there is nothing to prove) and fix n > 1. We can and do assume that IIEz=,Yk lip < co, and then, since
Ii Ii
where Xn* =
max X,, l
we infer that
By Holder’s exceed
inequality
the last expression
does not
~~-,X~~~~~fll~~~~~,lX~l,:i”p
=n”4(j,llYkll~]“p < n”Y
II II 5
Y,
k=l
p’
lip is finite as well. Put Sk = c:=1X,, 1 < k G n, and S,, = 0. By the Lagrange Theorem
lfc~=,xk
Applying our lemma to the sets Ak = { Xk > t }, Bk={Yk>t} we P(X,* > t)< see that 2P(Y,* > t). In particular, II Xz lip < 2 llY,* lip <
2 llC;=,Y, II/,. Thus
Therefore, and finally,
; i=l
328
(S,?’
- Sp_;‘) as desired.
Volume 9, Number
4
STATISTICS
AND PROBABILITY
April 1990
LETTERS
To see that the constant O(p) is best possible, consider random variables (X,) and (Y,) defined on the unit interval as follows: X, = I,, zmA), Y, = I[2-A,*-“+l), k=l, 2 ).... If &o={$, ‘52}, F”= u{ X ,,..., X,}, n 2 1, then (X,) and (Y,,) are tangent sequences of random variables, and, since the random variable 1Xx = LU,,~~~I~,~L, has exponential tail, 11CX, lip = O(p). On the other hand, since Y,‘s are disjointly supported, we have 0 IiCY, IIn = 1, which completes the proof.
Then:
We wish to conclude by remarking that our result provides a surprisingly simple proof of the convex function inequality (when specialized to the case of convex powers) (cf., e.g., Burkholder et al., 1972; Garsia, 1973).
~l~(~Y~l~,~~,~(~~((~Y~)~l~))‘”
Corollary. Let (X,) be a sequence of non-negative random variables adapted to the filtration (gn). Then, for 1 < p < co we have that
Proof. As was observed by Kwapien and Woyczyhski (1986), for every sequence (X,) of random variables there exists (perhaps on an enlarged probability space) a sequence (Y,) tangent to (X,), with the following additional property: There exists a u-field 9 such that: (a) the conditional distribution of Y, given 9 is the same as the conditional distribution of Y, given Sk_,, k>, 1; (b) (Y,) is a sequence of Sconditionally independent random variables. Now, given a sequence (X,), let as in the corollary, (Y,) and 9 satisfy (a) and (b) above.
and by Jensen’s inequality for conditional tations and our result we obtain:
expec-
=I/cykllyG3PllcX/lp9 which completes
the proof.
q
References Burkholder, D.L. (1973) Distribution function inequalities for martingales, Ann. Probab. 1, 19-42. Burkholder, D.L., B.J. Davis and R.F. Gundy (1972) Integral inequalities for convex functions of operators on martingales, in: Proc. Sixth Berkeley Symp. Math. Smut. Probab., Vol. 2 (Univ. of California Press, Berkeley, CA) pp. 223-240. Garsia, A.M. (1973). Martingale Inequalities. Seminar Notes on Recent Progress (Benjamin, Reading, MA). Hitczenko, P. (1988), Comparison of moments for tangent sequences of random variables, Probab. TheoN Rel. Fields 78, 223-230. Kwapieh, S. and W.A. Woyczyhski (1986). Semimartingale integrals via decoupling inequalities and tangent processes. Case Western Reserve University Preprint, 86#56. Zinn, J. (1985). Comparison of martingale differences, in: A. Beck, R. Dudley, M. Hahn, J. Kuelbs and M. Marcus. eds.. Probability in Banach Spaces V (Springer, Berlin, Heidelberg, New York) pp. 453-457.
329
Letters 9 (1990) 327-329
April 1990
BEST CONSTANT IN THE DECOUPLING RANDOM VARIABLES
Pawel HITCZENKO
INEQUALITY
FOR NON-NEGATIVE
*
Department of Mathematics, Texas A&M Unioersity, College Station, TX 77843, USA Received September 1988 Revised February 1989
Abstract: A simple proof of the following ~~CX4,~34CYkllp~
inequality
is given:
Pal,
where, for n > 1, X, and Y, are Fn-measurable non-negative 9 n-t. Our proof gives the best possible order of constant.
random
variables
with indentical
conditional
distributions,
given
AMS 1980 Subject Classifications: 60E15. Keywords: Non-negative
random
variables,
tangent
sequences,
In the present paper we obtain the best order of constant appearing in the comparison theorem for the L,-norms, p > 1, for sums of tangent sequences of non-negative random variables. Let (0, S-, P) be a probability space with the ascending sequence (Fn) of sub-u-fields of 9. Two (Sn)adapted sequences ( X,) and (Y,) of random variables are tangent if for each n >, 1 the conditional distributions of X, and Y, given pn_i coincide a.s., i.e. if for every real number t, P( X, > r ) .9_ 1) = P( Y, > t ) gn _ 1) a.s. The following conjecture of Kwapieh and Woyczynski (1986) was substantiated by the author in Hitczenko (1988): For every continuous increasing function @ : Iw + --f R!+ such that Q(O) = 0 and @(2t) G a@(t) for some constant (Y and all positive number t, there exists a constant c such that the inequality
holds for all tangent sequences (X,) and (Y,) of non-negative random variables. The proof was * On leave from Academy
of Physical
Education,
inequality.
based on Davis’s decomposition and the ‘good X inequality’ of Burkholder (1973). Our aim here is to give a much simpler proof of the special (but the most interesting) case Q(t) = tP, p 2 1, of the above inequality (for the concave powers, see Zinn, 1985; Hitczenko, 1988, Remark); a proof which gives the best possible growth rate of the constant. Theorem. Let 1 < p < co. Then for all tangent sequences (X,,) and (Y,,) of non-negative random variables the following is true:
where 11X lip denotes the L,-norm of a random variable X. The constant 3p is best possible (up to an absolute constant, not depending on p). We shall need the following lemma, which comes from Hitczenko (1988) and we include the proof for the convenience of the reader: Lemma. If A,, and B,, are 9”-measurable 52 such that
subsets of
Warsaw,
Poland. 0167-7152/90/$3.50
decoupling
0 1990, Elsevier Science Publishers
P(A,I~l-,)=P(B,I~_,)
a.~.,
n>l,
B.V. (North-Holland) 321
Volume
9. Number
STATISTICS
4
AND PROBABILITY
April 1990
LETTERS
then =p c E(S,P-’ i=l
P(UA,) < 2P(UB,). Proof. Denote by A’ the complement of a set A and put C,=D, C,,=B;n ...nBB,c_,, na2. Then
f’(UA,) < f’(W)
+ f’((U4,)
[=I
=piE(Sr-‘-S,T;‘)E Since
n (f-W;)). E
But NU4,)
- S,C;‘)
n (f-J%>) = P((U4)
n (nC,))
G P(U(4
n CR>>
G CP(4
n Cn>
=E
X,+ i X,+
= CE~(CM4)
which completes
E(XkIFA_,)e
f
E(YklFk_l)q
+:+;I+?), k=r+l
= CNCJ%%
13-I)
= CEI(Cn)f’@n
13-1)
= zP(C,,n
2 k=,+l
B,)= P(UBn),
the proof of the lemma.
q
Proof of the theorem. In our proof we shall use some ideas of Garsia (1973). Assume that p > 1 (since otherwise there is nothing to prove) and fix n > 1. We can and do assume that IIEz=,Yk lip < co, and then, since
Ii Ii
where Xn* =
max X,, l
we infer that
By Holder’s exceed
inequality
the last expression
does not
~~-,X~~~~~fll~~~~~,lX~l,:i”p
=n”4(j,llYkll~]“p < n”Y
II II 5
Y,
k=l
p’
lip is finite as well. Put Sk = c:=1X,, 1 < k G n, and S,, = 0. By the Lagrange Theorem
lfc~=,xk
Applying our lemma to the sets Ak = { Xk > t }, Bk={Yk>t} we P(X,* > t)< see that 2P(Y,* > t). In particular, II Xz lip < 2 llY,* lip <
2 llC;=,Y, II/,. Thus
Therefore, and finally,
; i=l
328
(S,?’
- Sp_;‘) as desired.
Volume 9, Number
4
STATISTICS
AND PROBABILITY
April 1990
LETTERS
To see that the constant O(p) is best possible, consider random variables (X,) and (Y,) defined on the unit interval as follows: X, = I,, zmA), Y, = I[2-A,*-“+l), k=l, 2 ).... If &o={$, ‘52}, F”= u{ X ,,..., X,}, n 2 1, then (X,) and (Y,,) are tangent sequences of random variables, and, since the random variable 1Xx = LU,,~~~I~,~L, has exponential tail, 11CX, lip = O(p). On the other hand, since Y,‘s are disjointly supported, we have 0 IiCY, IIn = 1, which completes the proof.
Then:
We wish to conclude by remarking that our result provides a surprisingly simple proof of the convex function inequality (when specialized to the case of convex powers) (cf., e.g., Burkholder et al., 1972; Garsia, 1973).
~l~(~Y~l~,~~,~(~~((~Y~)~l~))‘”
Corollary. Let (X,) be a sequence of non-negative random variables adapted to the filtration (gn). Then, for 1 < p < co we have that
Proof. As was observed by Kwapien and Woyczyhski (1986), for every sequence (X,) of random variables there exists (perhaps on an enlarged probability space) a sequence (Y,) tangent to (X,), with the following additional property: There exists a u-field 9 such that: (a) the conditional distribution of Y, given 9 is the same as the conditional distribution of Y, given Sk_,, k>, 1; (b) (Y,) is a sequence of Sconditionally independent random variables. Now, given a sequence (X,), let as in the corollary, (Y,) and 9 satisfy (a) and (b) above.
and by Jensen’s inequality for conditional tations and our result we obtain:
expec-
=I/cykllyG3PllcX/lp9 which completes
the proof.
q
References Burkholder, D.L. (1973) Distribution function inequalities for martingales, Ann. Probab. 1, 19-42. Burkholder, D.L., B.J. Davis and R.F. Gundy (1972) Integral inequalities for convex functions of operators on martingales, in: Proc. Sixth Berkeley Symp. Math. Smut. Probab., Vol. 2 (Univ. of California Press, Berkeley, CA) pp. 223-240. Garsia, A.M. (1973). Martingale Inequalities. Seminar Notes on Recent Progress (Benjamin, Reading, MA). Hitczenko, P. (1988), Comparison of moments for tangent sequences of random variables, Probab. TheoN Rel. Fields 78, 223-230. Kwapieh, S. and W.A. Woyczyhski (1986). Semimartingale integrals via decoupling inequalities and tangent processes. Case Western Reserve University Preprint, 86#56. Zinn, J. (1985). Comparison of martingale differences, in: A. Beck, R. Dudley, M. Hahn, J. Kuelbs and M. Marcus. eds.. Probability in Banach Spaces V (Springer, Berlin, Heidelberg, New York) pp. 453-457.
329