2 Li's Weak Correlation Inequality
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Wenbo V. Li’s Contributions
285
It is known that (2.1) holds for k D 1. There have been several claims of a proof
of the Gaussian correlation conjecture.
2.3 Li’s Comparison Theorem
Let n ; n 1 be i.i.d. standard normalP
random variables,
fan g and fbn g be sequences
P1
of strictly positive real numbers with 1
a
<
1,
nD1 n
nD1 bn < 1.
Theorem 2.3 (Li [8]) If
1
X
j1
an =bn j < 1;
(2.2)
nD1
then as ! 0
P
1
X
an
2
n
Ä
2
Á
nD1
1
Y
.bn =an /
Á1=2
P
nD1
1
X
bn
2
n
Ä
2
Á
:
(2.3)
nD1
Theorem 2 of Gao et al. [5] removes assumption (2.2) and shows (2.3) provided
Q1
nD1 .bn =an / < 1.
2.4 A Reversed Slepian Type Inequality
Let fXi ; 1 Ä i Ä ng and fYi ; 1 Ä i Ä ng be centered Gaussian random vectors. The
classical Slepian [15] inequality states that
If EXi2 D EYi2 and E.Xi Xj / Ä E.Yi Yj / for all 1 Ä i; j Ä n, then for any x
Á
Á
P max Xi Ä x Ä P max Yi Ä x
1ÄiÄn
1ÄiÄn
The Slepian inequality has played an important role in various probability
estimates for Gaussian measure. Li and Shao [13] established the following reversed
inequality, which is a special case of Theorem 2.2 in [13].
Theorem 2.4 (Li and Shao [13]) Assume EXi2 D EYi2 D 1 and 0 Ä E.Xi Xj / Ä
E.Yi Yj / for all 1 Ä i; j Ä n. Then for x 0
Á
Á
P max Xi Ä x Ä P max Yi Ä x
1ÄiÄn
Á
1ÄiÄn
Ä P max Xi Ä x
1ÄiÄn
Y
1Äi
2 arcsin.EXi Xj / Áexp
2 arcsin.EYi Yj /
x2 =.1CEYi Yj /
:
286
Q.-M. Shao
2.5 The First Exit Time of a Brownian Motion
from an Unbounded Convex Domain
Let B.t/ D .B1 .t/;
; Bd .t// 2 Rd ; t
0 be a standard d-dimensional Brownian
motion, where Bi .t/; 1 Ä i Ä d are independent standard Brownian motions. Let
D D f.x; y/ 2 RdC1 W y > f .x/; x 2 Rd g;
where f .x/ is a convex function on Rd . The first exit time D of a .dC1/-dimensional
Brownian motion from D starting at the point .x0 ; f .x0 / C 1/ is defined by
D
0 W .x0 C B.t/; f .x0 / C 1 C B0 .t// 62 Dg;
D infft
where B0 .t/ is a standard Brownian motion independent of B.t/. Bañuelos et al. [1]
proved that If d D 1, f .x/ D jxj2 , then
log P.
t1=3 as t ! 1:
t/
D
Li [10] gave a very general estimate for log P.
exp.kxkp /; p > 0;
lim t 1 .log t/2=p log P.
D
t!1
where v D .d
D
> t/. In particular, for f .x/ D
t/ D
j2v =2;
2/=2 and jv is the smallest positive zero of the Bessel function Jv .
2.6 Lower Tail Probabilities
Let fXt ; t 2 Tg be a real valued Gaussian process indexed by T with EXt D 0.
Lower tail probability refers to
P sup.Xt
t2T
Xt0 / Ä x
Á
as x ! 0; t0 2 T
or
Á
P sup Xt Ä x
as jTj ! 1:
t2T
Li and Shao [14] obtained a general result for the lower tail probability of nonstationary Gaussian process
P sup.Xt
t2T
Xt0 / Ä x
Á
as x ! 0;
Wenbo V. Li’s Contributions
287
Special cases include:
(a) Let fX.t/; t 2 Œ0; 1d g be a centered Gaussian process with X.0/ D 0 and
stationary increments, that is
X s /2 D
8 t; s 2 Œ0; 1d ; E.Xt
2
.kt
sk/:
If there are 0 < ˛ Ä ˇ < 1 such that .h/=h˛ is non-decreasing and .h/=hˇ
non-increasing, then as ! 0
ln P. sup X.t/ Ä . //
1
log :
t2Œ0;1d
(b) Let fX.t/; t 2 Œ0; 1d g be a centered Gaussian process with X.0/ D 0 and
E.Xt Xs / D
d
Y
1
iD1
2
.
2
2
.ti / C
.si /
2
.jti
si j//:
If there are 0 < ˛ Ä ˇ < 1 such that .h/=h˛ is non-decreasing and .h/=hˇ
non-increasing, then as ! 0, then as ! 0
ln P. sup X.t/ Ä
d
. //
1
lnd :
t2Œ0;1d
2.7 Large Deviations for Self-Interaction Local Times
Let B.t/; t
0 be a one-dimensional Brownian motion and k
interaction local time given by
2. The self-
Z
ˇt D
Œ0;tk
1fB.s1/DB.s2/D
DB.sk /g ds1
dsk
measures the intensity of the k-multiple self-intersection of the Brownian path. It is
known that
Z 1
ˇt D
Lk .t; x/dx;
1
where
Z
t
L.t; x/ D
0
is the local time of the Brownian motion.
ıx .B.s//ds
288
Q.-M. Shao
Chen and Li [3] proved that the following holds:
lim x
2=.k 1/
x!1
D
1
4.k
log P.ˇ1
x/
k C 1 Á.3
1/
2
k/=.k 1/
B
1
k
1 Á2
;
1 2
;
where B. ; / is the beta function.
This is a special case of Theorem 1.1 in [3].
2.8 Ten Lectures on Small Value Probabilities and Applications
Wenbo delivered ten comprehensive lectures on small value probabilities and
applications at NSF/CBMS Regional Research Conference in the Mathematical
Sciences, University of Alabama in Huntsville, June 04–08, 2012. We highly
recommend them to anyone who is interested in this topic.
3 Wenbo’s Open Problems
From time to time Wenbo raised many interesting open questions. In this section we
summarize a selection of them, some of which might not be originally due to him.
We refer to Wenbo’s ten lectures for details.
1. Gaussian products conjecture:
For any centered Gaussian vector .X1 ;
E.X12m
Xn2m /
; Xn /, it holds
E.X12m /
E.Xn2m /
for each integer m 1.
It is known it is true when m D 1 (Frenkel [4]) .
2. Gaussian minimum conjecture:
Let .Xi ; 1 Ä i Ä n/ be a centered Gaussian random vector. Then
E min jXi j
1ÄiÄn
E min jXi j;
1ÄiÄn
where Xi are independent Gaussian random variables with E.Xi 2 / D E.Xi2 /.
A weak result was proved by Gordon et al. [6]:
E min jXi j
1ÄiÄn
.1=2/E min jXi j:
1ÄiÄn
Wenbo V. Li’s Contributions
289
3. Conjecture: Let
P i ; 1 Ä i Ä n be i.i.d. r.v.’s P.
fai g satisfying niD1 a2i D 1
P j
n
X
ai i j Ä 1
Á
i
D ˙1/ D 1=2. Then for any
1=2:
iD1
The best known lower bound is 3=8.
In the following open questions 4–6, let ij be i.i.d. Bernoulli random variables
with P. ij D ˙1/ D 1=2.
4. Determinant of Bernoulli matrices:
Let Mn D . ij /n n . It is easy to show that
E.det.Mn /2 / D nŠ:
It was proved by Tao and Vu [16] that
P j det.Mn /j Ä
Á
p
nŠ exp. 29.n log n/1=2 / D o.1/:
Conjecture: For 0 < ı < 1,
p Á
ı/ nŠ D o.1/
P j det.Mn /j Ä .1
and with probability tending to 1
p
j det.Mn /j D nO.1/ nŠ:
5. Singularity probability of random Bernoulli matrices:
Let Mn D . ij /n n . Clearly, one has
P.det.Mn / D 0/
.1 C o.1//n2 21 n :
It is conjectured that
P.det.Mn / D 0/ D
Án
1
C o.1/ :
2
The best known result is due to Bourgain et al. [2]:
Án
1
P.det.Mn / D 0/ Ä p C o.1/ :
2
290
Q.-M. Shao
6. Gaussian Hadamard conjecture:
The Hadamard conjecture can be restated as
P
max j
1Äj6DkÄn
n
X
ij ik j
<1
Á
;
iD1
where n D 4m
The Gaussian Hadamard conjecture is: Let
random variables. Then
max j
ln P
n2
2
1Äj6DkÄn
n
X
ij ik j
<1
ij
be i.i.d. standard normal
Á
n2 :
iD1
7. The traveling salesman problem:
Let
Ln D min
n 1
X
jX
.iC1/
.i/ j
X
iD1
be the shortest tour of n i.i.d. uniform points fXi ; 1 Ä i Ä ng
denotes a permutation of f1;
; ng. It is known that
E.Ln /=n.d
1/=d
Œ0; 1d , where
! ˇ.d/:
Open question: What is the value of ˇ.d/? Does the central limit theorem
hold?
8. Two-sample matching:
Let fXi g and fYi g be i.i.d. uniformly distributed on Œ0; 12 . Consider
Mn D min
n
X
jXi
Y
.i/ j;
Mn D min max jXi
iD1
1ÄiÄn
Y
.i/ j:
It is known that there exist 0 < c0 < c1 < 1 such that
EMn
c0 Ä p
Ä c1 ; c0 Ä
n
n log n
EMn
1=2 .log n/3=4
Ä c1 :
Open
p question: What are the exact limits? What are the limiting distributions
of Mn = n log n and Mn =.n 1=2 .log n/3=4 / ?
Acknowledgements We would like to thank the referee for his/her helpful suggestions/comments.
This work was partly supported by Hong Kong RGC GRF 403513 and 14302515.
Wenbo V. Li’s Contributions
291
References
1. R. Banuelos, R. DeBlassie, R. Smits, The first exit time of Brownian motion from interior of a
parabola. Ann. Probab. 29, 882–901 (2001)
2. J. Bourgain, V. Vu, P.M. Wood, On the singularity probability of discrete random matrices. J.
Funct. Anal. 258, 559–603 (2010)
3. X. Chen, W.V. Li, Large and moderate deviations for intersection local times. Probab. Theory
Relat. Fields 128, 213–254 (2004)
4. P.E. Frenkel, Pfaffians, Hafnians and products of real linear functionals. Math. Res. Lett. 15,
351–358 (2008)
5. F. Gao, J. Hannig, T.Y. Lee, F. Torcaso, Exact L2 small balls of Gaussian processes. J. Theor.
Probab. 17, 503–520 (2004)
6. Y. Gordon, A. Litvak, C. Schtt, E. Werner, On the minimum of several random varaibles. Proc.
Am. Math. Soc. 134, 3665–3675 (2006)
7. J. Kuelbs, W.V. Li, Metric entropy and the small ball problem for Gaussian measures. J. Funct.
Anal. 116, 133–157 (1993)
8. W.V. Li, Comparison results for the lower tail of Gaussian seminorms. J. Theor. Probab. 5,
1–31 (1992)
9. W.V. Li, A Gaussian correlation inequality and its applications to small ball probabilities.
Electron. Commun. Probab. 4, 111–118 (1999)
10. W.V. Li, The first exit time of a Brownian motion from an unbounded convex domain. Ann.
Probab. 31, 1078–1096 (2003)
11. W.V. Li, Ten Lectures on Small Deviation Probabilities: Theory and Applications (2012), http:
jamesyli.com/wenboli_backup
12. W.V. Li, W. Linde, Approximation, metric entropy and small ball estimates for Gaussian
measures. Ann. Probab. 27, 1556–1578 (1999)
13. W.V. Li, Q.M. Shao, A normal comparison inequality and its applications. Probab. Theory
Relat. Fields 122, 494–508 (2002)
14. W.V. Li, Q.M. Shao, Lower tail probabilities for Gaussian processes. Ann. Probab. 32, 216–242
(2004)
15. D. Slepian, The one-sided barrier problem for Gaussian noise. Bell Syst. Tech. J. 41, 463–501
(1962)
16. T. Tao, V. Vu, On random ˙1 matrices: singularity and determinant. Random Struct.
Algorithm. 28, 1–23 (2006)
Part III
Stochastic Processes
Orlicz Integrability of Additive Functionals of
Harris Ergodic Markov Chains
Radosław Adamczak and Witold Bednorz
Abstract For a Harris ergodic Markov chain .Xn /n 0 , on a general state space,
started from the small measure or from the stationary
distribution, we provide
P
optimal estimates for Orlicz norms of sums
f
.X
is the first
i /, where
iD0
regeneration time of the chain. The estimates are expressed in terms of other
Orlicz norms of the function f (with respect to the stationary distribution) and the
regeneration time (with respect to the small measure). We provide applications
to tail estimates for additive functionals of the chain .Xn / generated by unbounded
functions as well as to classical limit theorems (CLT, LIL, Berry-Esseen).
Keywords Limit theorems • Markov chains • Orlicz spaces • Tail inequalities •
Young functions
Mathematics Subject Classification (2010). Primary 60J05, 60E15; Secondary
60K05, 60F05
1 Introduction and Notation
Consider a Polish space X with the Borel -field B and let .Xn /n 0 be a time
homogeneous Markov chain on X with a transition function PW X B ! Œ0; 1.
Throughout the article we will assume that the chain is Harris ergodic, i.e. that there
exists a unique probability measure on .X ; B/ such that
kPn .x; /
kTV ! 0
for all x 2 X , where k kTV denotes the total variation norm, i.e. k kTV D
supA2B j .A/j for any signed measure .
R. Adamczak ( ) • W. Bednorz
Institute of Mathematics, University of Warsaw, Banacha 2, 02–097 Warszawa, Poland
e-mail: radamcz@mimuw.edu.pl; 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_13
295
296
R. Adamczak and W. Bednorz
One of the best known and most efficient tools of studying such chains is the
regeneration technique [4, 30], which we briefly recall bellow. We refer the reader
to the monographs [28, 31] and [12] for extensive description of this method and
restrict ourselves to the basics which we will need to formulate and prove our results.
One can show that under the above assumptions there exists a set (usually called
small set) C 2 E C D fA 2 BW .A/ > 0g, a positive integer m, ı > 0 and a Borell
probability measure on X (small measure) such that
Pm .x; /
ı ./
(1.1)
for all x 2 C. Moreover one can always choose m and in such a way that .C/ > 0.
Existence of the above objects allows to redefine the chain (possibly on an
enlarged probability space) together with an auxiliary regeneration structure. More
precisely, one defines the sequence .XQ n /n 0 and a sequence .Yn /n 0 of f0; 1g-random
variables by requiring that XQ 0 have the same distribution as X0 and specifying
XQ
the conditional probabilities (see [28, Chap. 17.3.1]) as follows. Denote Fkm
D
Y
..XQ i /iÄkm / and Fk 1 D ..Yi /iÄk 1 /. For x 2 C let
r.x; y/ D
ı .dy/
:
Pm .x; dy/
Note that the density in the definition of r is well-defined by (1.1) and the RadonNikodym theorem. Moreover it does not exceed ı 1 and so r.x; y/ Ä 1. Now for
A1 ; : : : ; Am 2 B set
P fYk D 1g \
m
\
Q
X
fXQ kmCi 2 Ai gjFkm
; FkY 1 ; XQ km D x
Á
iD1
D P fY0 D 1g \
Z
Z
m
\
fXQ i 2 Ai gjXQ 0 D x
Á
iD1
r.x; xm /P.xm 1 ; dxm /P.xm 2 ; dxm 1 /
D 1fx2Cg
A1
P.x; dx1 /
Am
and
P.fYk D 0g \
m
\
Q
X
fXQ kmCi 2 Ai gjFkm
; FkY 1 ; XQ km D x/
iD1
D P.fY0 D 0g \
Z
fXQ i 2 Ai gjXQ 0 D x/
iD1
Z
1
D
A1
m
\
Am
Á
1fx2Cg r.x; xm / P.xm 1 ; dxm /
P.x; dx1 /:
Orlicz Integrability of Additive Functionals of Markov Chains
297
Note that if XQ km D x … C, then (conditionally) almost surely Yk D 0 and the
XQ
conditional distribution of .XQ kmC1 ; : : : ; XQ .kC1/m / given Fkm
; FkY 1 ; XQ km is the same as
the conditional distribution of .XkmC1 ; : : : ; X.kC1/m / given Xkm D x.
The process .XQ n ; Yn /n 0 is usually referred to as the split chain (although some
authors reserve this name for the process .XQ nm ; Yn /n 0 ). In the special case of m D 1
the above construction admits a nice ‘algorithmic’ interpretation: if XQ n D x 2 C,
then one tosses a coin with probability of heads equal to ı; if one gets heads, then
the point XQ nC1 is generated from the measure (which is independent of x) and one
sets Yn D 1, otherwise the new point XQ nC1 is generated from the transition function
.P.x; / ı . //=.1 ı/ and one sets Yn D 0; if XQ n D x … C, then Yn D 0 and
XQ nC1 is generated from P.x; /. In the general case this interpretation works for the
process .XQ nm ; Yn / and the above formulas allow to fill in the missing values of XQ nmCi
in a consistent way.
One can easily check that .XQ n /n 0 has the same distribution as .Xn /n 0 and so we
may and will identify the two sequences (we will suppress the tilde). The auxiliary
variables Yn can be used to introduce some independence which allows to recover
many results for Markov chains from corresponding statements for the independent
(or one-dependent) case. Indeed, observe that if we define the stopping times
.0/ D inffk
0; Yk D 1g; .i/ D inffk > .i
1/W Yk D 1g; i D 1; 2; : : : ;
then the blocks
R0 D .X0 ; : : : ; X
.0/mCm 1 /; Ri
D .Xm.
.i 1/C1/ ; : : : ; Xm .i/Cm 1 /
are one-dependent, i.e. for all k, .Ri ; i < k/ is independent of .Ri ; i > k/. In
the special case, when m D 1 (the strongly aperiodic case) the blocks Ri are
independent. Moreover, for i 1 the blocks Ri form a stationary sequence.
PnIn particular for any function f W X ! R, the corresponding additive functional
iD0 f .Xi / can be split (modulo the initial and final segment) into a sum (of random
length) of one-dependent (independent for m D 1) identically distributed summands
m .iC1/Cm 1
X
si . f / D
f .Xj /:
jDm. .i/C1/
A crucial and very useful fact is the following equality, which follows from
Pitman’s occupation measure formula ([35, 36], see also Theorem 10.0.1 in [28]),
i.e. for any measurable FW X f0; 1g ! R,
E
.0/
X
iD0
F.Xmi ; Yi / D ı
1
.C/ 1 E F.X0 ; Y0 /;
(1.2)