Appendix. Some Generalities on Orlicz Young Functions and Orlicz Spaces
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324
R. Adamczak and W. Bednorz
(iii) There exist positive constants C; x0 such that
F.sx/
F.x/
for all s
1, x
1 G.sx/
C
G.x/
x0 .
Lemma 22 For any Young function
x 0,
such that limx!1 .x/=x D 1 and any
/ 1 .x/
xÄ.
1
.x/ Ä 2x:
Moreover, the right hand side inequality holds for any strictly increasing continuous function W Œ0; 1/ ! Œ0; 1/, such that .0/ D 0, .1/ D 1,
limx!1 .x/=x D 1.
Lemma 23 Let ' and
be two Young functions. Assume that
lim '.x/=x D 1:
x!1
If '
1
ı
is equivalent to a Young function, then so is .
/
1
ı' .
Proof It is easy to see that under the assumptions of the lemma we also have
.x/ are finite for all x
0. Applying
limx!1 .x/=x D 1 and thus ' .x/,
1
Lemma 21 with F D ' 1 , G D
, we get that
'
for some C > 0, all s
enough,
1
.sx/
1 .sx/
C
1
'
1
.x/
1 .x/
1 and x large enough. By Lemma 22 we obtain for x large
. / 1 .sx/
.' / 1 .sx/
.4C/
/ 1 .x/
;
.' / 1 .x/
1.
which ends the proof by another application of Lemma 21.
t
u
Acknowledgements Research partially supported by MNiSW Grant N N201 608740 and the
Foundation for Polish Science.
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Bounds for Stochastic Processes on Product
Index Spaces
Witold Bednorz
Abstract In this paper we discuss the question of how to bound the supremum of
a stochastic process with an index set of a product type. It is tempting to approach
the question by analyzing the process on each of the marginal index sets separately.
However it turns out that it is necessary to also study suitable partitions of the entire
index set. We show what can be done in this direction and how to use the method to
reprove some known results. In particular we point out that all known applications of
the Bernoulli Theorem can be obtained in this way. Moreover we use the shattering
dimension to slightly extend the application to VC classes. We also show some
application to the regularity of paths of processes which take values in vector spaces.
Finally we give a short proof of the Mendelson–Paouris result on sums of squares
for empirical processes.
Keywords Shattering dimension • Stochastic inequalities • VC classes
Mathematics Subject Classification (2010). Primary 60G15; Secondary 60G17
1 Introduction
In this paper I denotes a countable set and .F; k k/ a separable Banach space.
Consider the class A of subsets of I. We say that the class A satisfies the maximal
inequality if for any symmetric independent random variables Xi , i 2 I taking values
Research partially supported by MNiSW Grant N N201 608740 and MNiSW program Mobility
Plus.
W. Bednorz ( )
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
e-mail: wbednorz@mimuw.edu.pl
© Springer International Publishing Switzerland 2016
C. Houdré et al. (eds.), High Dimensional Probability VII,
Progress in Probability 71, DOI 10.1007/978-3-319-40519-3_14
327
328
W. Bednorz
in F the following inequality holds
E sup
A2A
X
Xi 6 KE
i2A
X
Xi ;
(1.1)
i2I
where K depends on A only. We point out that in this paper K will be used to denote
constants that appear in the formulation of our results and may depend on them.
We use c; C; L; M to denote absolute constants, which may change their values from
line to line by numerical factors. Also we write to express that two quantities are
comparable up to a universal constant. This will help us to reduce the notation in
this paper. It is an easy observation to see that (1.1) is equivalent to
E sup
A2A
X
vi "i 6 KE
i2A
X
vi "i ;
(1.2)
i2I
where .vi /i2I , consists of vectors in F and ."i /i2I is a Bernoulli sequence, i.e. a
sequence of independent r.v.’s such that P."i D ˙1/ D 12 .
To understand what is the proper characterization of such classes A we recall
here the notion of VC dimension. We say that A has VC dimension d if there exists
a set B
I, jBj D d such that jfB \ A W A 2 Agj D 2d but for all B
I,
jBj > d, jfB \ A W A 2 Agj < 2dC1 . It means that A shatters some set B of
cardinality d, but does not shatter any set of cardinality d C 1. The result which
has been proved in [1] as a corollary of the Bernoulli Theorem states that finite
VC dimension is the necessary and sufficient condition for the class A to have the
property (1.1). Since our paper refers often to the Bernoulli Theorem we recall its
formulation. We begin by mentioning Talagrand’s result for Gaussian’s
P processes. In
order to find two-sided bounds for supremum of the process G.t/ D i2I ti gi , where
t 2 T
`2 .I/ and .gi /i2I is a Gaussian sequence, i.e. a sequence of independent
standard Gaussian r.v.’s we need Talagrand’s 2 .T/ numbers, cf. Definition 2.2.19
in [15] or (3.1) below. By the well known Theorem 2.4.1 in [15] we have
2 .T/:
E sup G.t/
t2T
(1.3)
The Bernoulli Theorem, i.e. Theorem 1.1 in [1], concerns a similar question for
processes of random signs.
Theorem 1.1 Suppose that T
E sup
X
t2T i2I
`2 .I/. Then
!
ti "i
inf
T T1 CT2
sup ktk1 C
t2T1
2 .T2 /
:
where the infimum is taken over all decompositions
T1 C T2 D ft1 C t2 W t1 2
P
T1 ; t2 2 T2 g that contain the set T and ktk1 D i2I jti j.
Bounds for Stochastic Processes on Product Index Spaces
329
Note that if 0 2 T then we can also require that 0 2 T2 in the above result. The
consequence of Theorem 1.1 to our problem with maximal inequalities is as follows.
Theorem 1.2 The class A satisfies (1.1) with a finite constant K if and only if A is
a VC class of a finite dimension. Moreover the square root of the dimension is up to
a universal constant comparable with the optimal value of K.
Observe that part of the result is obvious. Namely one can easily show that
if A satisfies the maximal inequality then it is necessarily a VC class of a finite
dimension. Indeed let ."i /i2I be a Bernoulli sequence. Suppose that set B
I is
shattered. Let xi D 1 for i 2 B and xi D 0, i 62 B. Obviously
ˇ
ˇ
ˇ
ˇ
ˇX ˇ p
ˇ
ˇX
ˇ
ˇ
ˇ
ˇ
Eˇ
xi " i ˇ D E ˇ
"i ˇ 6 jBj
ˇ
ˇ
ˇ
ˇ
i2I
i2B
and on the other hand
ˇ
ˇ
ˇ
ˇ
ˇX
ˇ
ˇX
ˇ
X
ˇ
ˇ
ˇ
ˇ
E sup ˇ
xi "i ˇ D E sup ˇ
xi " i ˇ > E
"i 1"i D1 D jBj=2:
ˇ
ˇ
A2A ˇ
A2A ˇ
i2A
i2A\B
i2B
p
Consequently if (1.1) holds then K > jBj=2. Therefore (1.1) implies that the
cardinality of B must be smaller or equal 4K 2 .
Much more difficult is to prove the converse statement, i.e. that for
p each VC
class A of dimension d inequality (1.2) holds with K comparable with d. In order
to prove this result one has to first replace the basic formulation of the maximal
inequality—(1.2) by its equivalent version
ˇ
ˇ
ˇ
ˇ
ˇX ˇ
ˇX ˇ
ˇ
ˇ
ˇ
ˇ
E sup sup ˇ
ti "i ˇ 6 KE sup ˇ
ti "i ˇ ;
ˇ
ˇ
ˇ
ˇ
A2A t2T
t2T
i2A
(1.4)
i2I
`2 .I/. Note that we use
where ."i /i2I is a Bernoulli sequence and 0 2 T
absolute values since part of our work concerns complex
spaces. However it is
P
important P
to mention that in the real case E supt2T i2I ti "i is comparable with
E supt2T j i2I ti "i j if 0 2 T and therefore we often require in this paper that
0 2 T `2 .I/. Let us denote
ˇ
ˇ
ˇ
ˇ
ˇX ˇ
ˇX ˇ
ˇ
ˇ
ˇ
ˇ
b.T/ D E sup ˇ
ti "i ˇ ; g.T/ D E sup ˇ
ti gi ˇ ;
ˇ
ˇ
ˇ
ˇ
t2T
t2T
i2I
i2I
where ."i /i2I , .gi /i2I are respectively Bernoulli and Gaussian sequence. We recall
that, what was known for a long time [5, 7], (1.4) holds when Bernoulli random
330
W. Bednorz
variables are replaced by Gaussians, i.e.
ˇ
ˇ
ˇ
ˇ
ˇX ˇ
ˇX ˇ
p
p
ˇ
ˇ
ˇ
ˇ
E sup sup ˇ
gi ti ˇ 6 C dE sup ˇ
ti gi ˇ D C dg.T/;
ˇ
ˇ
t2T ˇ
A2A t2T ˇ
i2A
(1.5)
i2I
for any 0 2 T
`2 .I/. Due to Theorem 1.1 one can cover the set T by T1 C T2 ,
where 0 2 T2 and
(
)
max sup ktk1 ; g.T2 / 6 Lb.T/:
(1.6)
t2T1
Therefore using (1.5) and (1.6)
ˇ
ˇ
ˇX ˇ
ˇ
ˇ
"i ti ˇ
E sup sup ˇ
ˇ
ˇ
A2A t2T
i2A
6 sup ktk1 C E sup sup j
t2T1
A2A t2T2
r
6 sup ktk1 C
t2T1
t2T1
2
2
E sup sup j
r
6 sup ktk1 C
2
t2T1
"i ti j
i2A
E sup sup j
r
6 sup ktk1 C
X
A2A t2T2
A2A t2T2
X
ti "i Ejgi jj
i2A
X
ti gi j
i2A
p
g.T2 / 6 CL db.T/:
This proves Theorem 1.2.
Here is another example in which a similar approach works. Let G be a compact
Abelian group and .vi /i2I a sequence of vectors taking values in F. Let i , i 2 I be
characters on G. A deep result of Fernique [4] is
E sup
h2G
X
i2I
vi i .h/gi 6 C E
X
i2I
ˇ
ˇ!
ˇX
ˇ
ˇ
ˇ
vi gi C sup E sup ˇ
x .vi / i .h/gi ˇ :
ˇ
ˇ
kx k61 h2G
i2I
Bounds for Stochastic Processes on Product Index Spaces
331
This can be rewritten similarly as (1.5), i.e. for any 0 2 T
complex space in this case)
`2 .I/ (which is a
ˇ
ˇ
ˇ
ˇ!
ˇX
ˇ
ˇX
ˇ
ˇ
ˇ
ˇ
ˇ
E sup sup ˇ
ti i .h/gi ˇ 6 C g.T/ C sup E sup ˇ
ti i .h/gi ˇ :
ˇ
ˇ
h2G t2T ˇ
t2T
h2G ˇ
i2I
(1.7)
i2I
Once again the Bernoulli Theorem permits us to prove a similar result for Bernoulli
sequences. Namely by Theorem 1.1 we get the decomposition T T1 C T2 , 0 2 T2
such that
)
(
max sup ktk1 ; g.T2 / 6 Lb.T/:
(1.8)
t2T1
Consequently using (1.7), (1.8) and j i .h/j 6 1 we get
ˇ
ˇ
ˇX
ˇ
ˇ
ˇ
E sup sup ˇ
ti i .h/"i ˇ
ˇ
ˇ
h2G t2T i2I
ˇ
ˇ
ˇ
ˇX
ˇ
ˇ
6 sup ktk1 C E sup sup ˇ
"i ti i .h/ˇ
ˇ
ˇ
t2T1
h2G t2T2 i2A
ˇ
ˇ
r
ˇX
ˇ
ˇ
ˇ
6 sup ktk1 C
E sup sup ˇ
ti i .h/gi ˇ
ˇ
2 h2G t2T2 ˇ i2I
t2T1
ˇ
ˇ!
ˇX
ˇ
ˇ
ˇ
6 sup ktk1 C C g.T2 / C sup E sup ˇ
ti i .h/gi ˇ
ˇ
ˇ
t2T1
t2T2
h2G i2I
!
X
ti i .h/gi j :
6 CL b.T/ C sup E sup j
t2T2
h2G
(1.9)
(1.10)
i2I
The final step is the Marcus–Pisier estimate [12] (see Theorem 3.2.12 in [15])
ˇ
ˇ
ˇ
ˇ
ˇX
ˇ
ˇX
ˇ
ˇ
ˇ
ˇ
ˇ
sup E sup ˇ
ti i .h/gi ˇ 6 M sup E sup ˇ
ti i .h/"i ˇ :
ˇ
ˇ
t2T2
h2G ˇ
t2T2
h2G ˇ
i2I
(1.11)
i2I
Note that (1.11) is deeply based on the translational invariance of the distance
dt .g; h/ D
X
i2I
2
jti j j i .g/
i .h/j
2
! 12
g; h 2 G:
(1.12)
332
W. Bednorz
Since we may assume that T2
T
T1 we get
ˇ
ˇ
ˇX
ˇ
ˇ
ˇ
sup E sup ˇ
ti i .h/"i ˇ
ˇ
t2T2
h2G ˇ i2I
ˇ
ˇ
ˇ
ˇ
ˇX
ˇ
ˇX
ˇ
ˇ
ˇ
ˇ
ˇ
ti i .h/"i ˇ C sup E sup ˇ
ti i .h/"i ˇ
6 sup E sup ˇ
ˇ t2T1 h2G ˇ
ˇ
t2T
h2G ˇ i2I
i2I
ˇ
ˇ
ˇX
ˇ
ˇ
ˇ
6 sup E sup ˇ
ti i .h/"i ˇ C Lb.T/:
ˇ
ˇ
t2T
h2G
(1.13)
i2I
Combining (1.9) with (1.11) and (1.13) we get the following result.
Theorem 1.3 Suppose that 0 2 T
`2 .I/. For any compact group G and a
collection of vectors vi 2 F in a complex Banach space .F; k k/ and characters i
on G the following holds
ˇ
ˇ
ˇ
ˇ!
ˇX
ˇ
ˇX
ˇ
ˇ
ˇ
ˇ
ˇ
ti i .h/"i ˇ 6 K b.T/ C sup E sup ˇ
ti i .h/"i ˇ :
E sup sup ˇ
ˇ
ˇ
ˇ
ˇ
h2G t2T
t2T
h2G
i2I
i2I
The aim of this note is to explore the questions described above in a unified
language. We consider random processes X.u; t/, .u; t/ 2 U T with values in
R or C, which means that we study stochastic processes defined on product index
sets. In particular we cover all canonical processes in this way. Indeed, suppose that
U RI or CI and T RI or CI are such that for any u 2 U and t 2 T we have that
P
2
i2I jui ti j < 1. Then for any family of independent random variables Xi such that
EXi D 0, EjXi j2 D 1,
X.u; t/ D
X
ui ti Xi ; u 2 U; t 2 T
i2I
is a well defined process. As we have already mentioned, our main class of examples
includes Gaussian canonical processes, where Xi D gi , i 2 I are standard normal
variables and Bernoulli canonical processes, where Xi D
P"i , i 2 I are random signs.
In particular, our goal is to findP
bounds for E supu2U k i2I ui vi "i k, where vi 2 F,
i 2 I, formulated in terms of Ek i2I vi "i k. One of our results is an application of the
shattering dimension introduced by Mendelson and Vershynin [14], which enables
us to generalize Theorem 1.2. In this way we deduce that under mild conditions on
U RI we have
E sup
u2U
X
i2I
ui vi Xi 6 KE
X
i2I
vi Xi
Bounds for Stochastic Processes on Product Index Spaces
333
for any independent symmetric r.v.’s Xi , i 2 I. We show how to apply the result to
the analysis of convex bodies and their volume in high dimensional spaces. On the
other hand we can use our approach to study processes X.t/ D .Xi .t//i2I , t 2 Œ0; 1,
which take values in RI or CI . For example to check whether paths t ! X.t/ belong
to `2 we should consider
X
X.u; t/ D
ui Xi .t/; u D .ui /i2I 2 U; t 2 Œ0; 1;
i2I
P
2
where U is the unit ball in `2 .I/, i.e. U D fu 2 RI W
i2I jui j 6 1g. The finiteness
of kX.t/k2 < 1 is equivalent to the finiteness of supu2U jX.u; t/j. Similarly we can
treat a well known question in the theory of empirical processes. Suppose that .E; B/
is a measurable space and F a countable family of measurable real functions on E.
Let X1 ; X2 ; : : : ; XN be independent random variables, which take values in .E; B/,
we may define
X.u; f / D
N
X
ui f .Xi /; u D .ui /NiD1 2 U; f 2 F ;
iD1
where U D BN .0; 1/ D fu 2 RN W
PN
sup jX.u; f /j2 D
u2U
iD1
N
X
jui j2 6 1g. Then it is clear that
jf .Xi /j2 ; for all f 2 F :
iD1
In the last section we give a short proof of Mendelson–Paouris result [13] that
provides an upper bound for E supu2U supf 2F jX.u; t/j.
2 Upper Bounds
For the sake of exposition we shall give an idea how to bound stochastic processes.
The approach we present slightly extends results of Latala [9, 10] and Mendelson–
Paouris [13]. Suppose that EjX.t/ X.s/j < 1 for all s; t 2 T. For each s; t 2 T
and n > 0 we define qN n .s; t/ as the smallest q > 0 such that
Fq;n .s; t/ D Eq 1 .jX.t/ X.s/j q/C
Z 1
D
P.jX.t/ X.s/j > qt/dt 6 Nn 1 :
(2.1)
1
We shall prove the following observation.
‘Note that the theory we describe below can be extended to the case of arbitrary
increasing numbers Nn .’ but for our purposes it is better to work in the type
n
exponential case where Nn D 22 for n > 0 and N0 D 1.
334
W. Bednorz
Lemma 2.1 Function qN n .s; t/, s; t 2 T is a distance on T, namely is symmetric,
satisfies the triangle inequality and qN n .s; t/ D 0 if and only if X.s/ D X.t/ a.s.
Proof Obviously qN n .s; t/ is finite and symmetric qN n .s; t/ D qN n .t; s/. To see that it
equals 0 if and only if P.jX.t/ X.s/j > 0/ > 0 note that if X.s/ Ô X.t/ then
EjX.t/ X.s/j > 0. The function q ! Fq;n .s; t/ is decreasing continuous and a.s.
Fq;n .s; t/ ! 1 if q ! 0 and EjX.t/ X.s/j > 0. Moreover Fq;n .s; t/ is strictly
decreasing on the interval fq > 0 W Fq;n .s; t/ > 0g and consequently qN .s; t/ is the
unique solution of the equation Fq;n .s; t/ D Nn 1 , namely
E.Nqn .s; t// 1 .jX.t/
X.s/j
qN n .s; t//C D Nn 1 :
Finally we show that qN satisfies the triangle inequality. Indeed for any u; v; w 2 T
either qN .u; v/ D 0 or qN .v; w/ D 0 or qN .u; w/ D 0 and the inequality is trivial or all
the quantities are positive and then
FqN n .u;v/;n .u; v/ D FqN n .v;w/;n .v; w/ D FqN n .u;w/;n .u; w/ D Nn 1 :
It suffices to observe
1
E .jX.u/ X.w/j qN n .u; v/
qN n .u; v/ C qN n .v; w/
Ã
Â
jX.u/ X.w/j C jX.w/ X.v/j
1
6E
:
qN n .u; v/ C qN n .w; v/
C
The function x ! .x
qN n .w; v//C
1/C is convex which implies that
.px C qy
1/C 6 p.x
1/C C q.y
1/C
for p; q > 0, p C q D 1 and x; y > 0. We use (2.2) for
xD
jX.v/ X.w/j
jX.u/ X.v/j
; yD
qN n .u; v/
qN n .v; w/
and
pD
qN n .v; w/
qN n .u; v/
; qD
:
qN n .u; v/ C qN n .w; v/
qN n .u; v/ C qN n .w; v/
Therefore
1
E .jX.u/ X.w/j qN n .u; v/ qN n .w; v//C
qN n .u; v/ C qN n .v; w/
Ã
Ã
Â
Â
jX.u/ X.v/j
jX.v/ X.w/j
1
1
6 pE
C qE
qN n .u; v/
qN n .v; w/
C
C
6 pNn 1 C qNn 1 D Nn 1
(2.2)