Transportation cost-information and concentration inequalities for bifurcating Markov chains

0.26 MB·English

by  Siméon Valère Bitseki Penda

Transportation cost-information and concentration
5
1 inequalities for bifurcating Markov chains
0
2
n S. Val`ere Bitseki Penda∗ Mikael Escobar-Bach †
a
J Arnaud Guillin ‡

7
2 January 28, 2015

]
R
P
.
h Abstract
t
a We investigate the transportation cost-information inequalities for
m
bifurcating Markov chains which are a class of processes indexed by
[ binary tree. These processes provide models for cell growth when

each individual in one generation gives birth to two offsprings in the
1
v next one. Transportation cost inequalities provide useful concentra-
3 tion inequalities. We also study deviation inequalities for the empiri-
9 cal means under relaxed assumptions on the Wasserstein contraction
6
of the Markov kernels. Applications to bifurcating non linear autore-
6
gressive processes are considered: deviation inequalities for pointwise
0
. estimates of the non linear leading functions.
1
0
Keywords: Transportation cost-information inequalities, Wasserstein distance, bi-
5
1 furcating Markov chains, deviation inequalities, geometric ergodicity.
:
v
i
X 1 Introduction
r
a
Roughly speaking, a bifurcating Markov chain is a Markov chain indexed
by a binary regular tree. This class of processes are well adapted for the
study of populations where each individual in one generation gives birth
∗CMAP, Ecole polytechnique route de Saclay, 91128, Palaiseau, France. email: bit-
[email protected]
†DepartmentofMathematicsandComputerScience,UniversityofSouthernDenmark,
Campus 55, 5230 Odense M, Denmark email:[email protected]
‡InstitutUniversitairedeFranceetLaboratoiredeMath´ematiques,CNRSUMR6620,
Universit´eBlaisePascal,24avenuedesLandais,BP80026,63177Aubi`ere,France. email:
[email protected]
1
to two offsprings in the next one. They were introduced by Guyon [29]
in order to study the Escherichia coli aging process. Namely, when a cell
divides into two offsprings, are the genetical traits identical for the two
daughter cells? Recently, several models of bifurcating Markov chains, or
models using the theory of bifurcating Markov chains, for example under
the form of bifurcating autoregressive processes, have been studied [1, 2, 29,
21, 18], showing that these processes are of great importance to analysis
of cell division. There is now an important literature covering asymptotic
theorems for bifurcating Markov chains such as Law of Large Numbers,
Central Limit Theorems, Moderate Deviation Principle, Law of Iterated
Logarithm, see for example [29, 30, 6, 19, 15, 18, 8] for recent references.
These limit theorems are particularly useful when applied to the statistics
of the bifurcating processes, enabling to provide efficient tests to assert if
the aging of the cell is different for the two offsprings (see [30] for real
case study). Of course, these limit theorems may be considered only in the
”ergodic” case, i.e. when the law of the random lineage chain has an unique
invariant measure.
However, limit theorems are only asymptotical results and one is often
faced to study only datas with a size limited population. It is thus very
natural to control the statistics non asymptotically. Such deviation inequal-
ities (or concentration inequalities) have been recently the subject of many
studies and we refer to the books of Ledoux [31] and Massart [35] for nice
introductions on the subject, developing both i.i.d. case and dependent
case with a wide variety of tools (Laplace controls, functional inequalities,
Efron-Stein,...). It was one of the goal of Bitseki et al. [8] to investigate
deviation inequalities for additive functionals of bifurcating Markov chain.
In their work, one of the main hypothesis is that the Markov chain as-
sociated to a random lineage of the population is uniformly geometrically
ergodic. It is clearly a very strong assumption, nearly reducing interesting
models to the compact case. The purpose of this paper is to considerably
weaken this hypothesis. More specifically, our aim is to obtain deviation
inequalities for bifurcating Markov chain when the auxiliary Markov chains
may satisfy some contraction properties in Wasserstein distance, and some
(uniform) integrabilty property. This will be done with the help of trans-
portation cost-information inequalities and directLaplace controls. In order
to present our result, we may now define properly the model of bifurcating
Markov chains.
2
1.1 Bifurcating Markov chains
First we introduce some useful notations. Let T be a regular binary tree in
which each vertex is seen as a positive integer different from 0. For r N,
∈
let
r
G = 2r,2r +1, ,2r+1 1 , T = G ,
r r q
··· −
n o q[=0
which denote respectively the r-th column and the first (r+1) columns of
the tree. The whole tree is thus defined by
∞
T = G .
r
r=0
[
A column of a given vertex n is G with r = log n , where x denotes
rn n ⌊ 2 ⌋ ⌊ ⌋
the integer part of the real number x.
Inthesequel,wewillseeTasagivenpopulationinwhicheachindividual
inonegenerationgivesbirthtotwooffspringsinthenextone. Thiswillmake
easier the introduction of different notions. The vertex n will denote the
individual n and the ancestor of individuals 2n and 2n+1. The individuals
whobelongto2N(resp. 2N+1)willbecalledindividualoftype0(resp. type
1). The column G and the first (r+1) columns T will denote respectively
r r
the r-th generation and the first (r+1) generations. The initial individual
will be denoted 1.
For each individual n, we look into a random variable X , defined on
n
a probability space (Ω, ,P) and which takes its values in a metric space
F
(E,d) endowed with its Borel σ-algebra . We assume that each pair of
E
random variables (X ,X ) depends of the past values (X ,m T )
2n 2n+1 m ∈ rn
only through X . In order to describe this dependance, let us introduce the
n
following notion.
Definition 1.1 (T-transition probability, see ([29])). We call T-transition
probability any mapping P :E 2 [0,1] such that
×E →
P(,A) is measurable for all A 2,
• · ∈ E
P(x, ) is a probability measure on (E2, 2) for all x E.
• · E ∈
In particular, for all x,y,z E, P(x,dy,dz) denotes the probability
∈
that the couple of the quantities associated with the children are in the
neighbourhood of y and z given that the quantity associated with their
mother is x.
3
For a T-transition probability P on E 2, we denote by P , P the first
0 1
×E
and the second marginal of P, that is P (x,A) = P(x,A E), P (x,A) =
0 1
×
P(x,E A) for all x E and A . Then, P (resp. P ) can be seen as
0 1
× ∈ ∈ E
the transition probability associated to individual of type 0 (resp. type 1).
For p 1, we denote by (Ep) (resp. (Ep)), the set of all p-
b
≥ B B E
measurable (resp. p-measurable and bounded) mappings f : Ep R.
E →
For f (E3), we denote by Pf (E) the function
∈ B ∈ B
x Pf(x)= f(x,y,z)P(x,dy,dz), whenitisdefined.
7→ ZS2
We are now in position to give a precise definition of bifurcating Markov
chain.
Definition 1.2 (Bifurcating Markov Chains, see ([29])). Let (X ,n T)
n
∈
be a family of E-valued random variables defined on a filtered probability
space (Ω, ,( ,r N),P). Let ν be a probability on (E, ) and P be a T-
r
F F ∈ E
transition probability. We say that (X ,n T) is a ( )-bifurcating Markov
n r
∈ F
chain with initial distribution ν and T-transition probability P if
X is -measurable for all n T,
• n Frn ∈
(X ) = ν,
1
• L
for all r N and for all family (f ,n G ) (E3)
n r b
• ∈ ∈ ⊆ B
E f (X ,X ,X ) = Pf (X ).
n n 2n 2n+1 r n n
F
"nY∈Gr (cid:12) # nY∈Gr
(cid:12)
(cid:12)
In the following, when unprecised, the filtration implicitly used will be
= σ(X ,i T ).
r i r
F ∈
Remark 1.3. WemayofcoursealsoconsiderinthisworkbifurcatingMarkov
chains on a a-ary tree (with a 2) with no additional technicalities, but
≥
heavy additional notations. In the same spirit, Markov chains of higher
order (such as BAR processes considered in [7]) could be handled by the
same techniques. A non trivial extension would be the case of bifurcating
Markov chains on a Galton-Watson tree (see for example [10] under very
strong assumptions), that we will consider elsewhere.
4
1.2 Transportation cost-information inequality
We recall that (E,d) is a metric space endowed with its Borel σ-algebra .
E
Given p 1, the Lp-Wasserstein distance between two probability measures
≥
µ and ν on E is defined by
1/p
Wd(ν,µ) = inf d(x,y)pdπ(x,y) ,
p
(cid:18)Z Z (cid:19)
where the infimum is taken over all probability measures π on the product
space E E with marginal distributions µ and ν (say, coupling of (µ,ν)).
×
This infimum is finite as soon as µ and ν have finite moments of order p.
When d(x,y) = 1 (the trivial measure), 2Wd(µ,ν) = µ ν , the
x6=y 1 k − kTV
total variation of µ ν.
−
The Kullback information (or relative entropy) of ν with respect to µ is
defined as
log dνdν, if ν µ
H(ν/µ) = dµ ≪
(+R else.
∞
Definition 1.4 (Lp-transportation cost-inequality). We say that the prob-
ability measure µ satisfies the Lp-transportation cost-information inequality
on (E,d) (and we write µ T (C)) if there is some constant C > 0 such
p
∈
that for any probability measure ν,
Wd(µ,ν) 2CH(ν/µ).
p ≤
This transportation inequality hapve been introduced by Marton [32, 33]
as a tool for (Gaussian) concentration of measure property. The follow-
ing result will be crucial in the sequel. It gives a characterization of L1-
transportation cost-inequality in term of concentration inequality. It is of
course one of the main tool to get deviation inequalities (via Markov in-
equality).
Theorem 1.5 ([11]). µ satisfies the L1-transportation cost-information in-
equality (say T ) on (E,d) with constant C > 0, that is, µ T (C), if and
1 1
∈
only if for any Lipschitzian function F :(E,d) R, F is µ-integrable and
→
λ2
exp(λ(F F ))dµ exp C F 2 λ R,
−h iµ ≤ 2 k kLip ∀ ∈
ZE (cid:18) (cid:19)
where F = Fdµ and
h iµ E
R F(x) F(y)
F = sup| − | < + .
Lip
k k d(x,y) ∞
x6=y
5
In particular, we have the concentration inequality
t2
µ(F F t) µ(F F t) exp t R.
−h iµ ≤ − ∨ −h iµ ≥ ≤ −2C F 2 ∀ ∈
k kLip!
In this work we will focus on transportation inequality T mainly. There
1
is now aconsiderable literature aroundthese transportation inequalities. As
aflavor, letuscite firstthecharacterization ofT asaGaussian integrability
1
property [20] (see also [23]).
Theorem 1.6 ([20]). µ satisfies the L1-transportation cost-information in-
equality (say T ) on (E,d) if and only if there exists δ > 0 and x E such
1 0
∈
that
µ eδd2(x,x0) < ,
∞
and the constant of the Transp(cid:16)ortation i(cid:17)nequality can be made explicit.
There is also a large deviations characterization [25]. Recent striking
results on transportation inequalities have been obtained for T , namely
2
that they are equivalent to dimension free Gaussian concentration [24], or
to a restricted class of logarithmic Sobolev inequalities [27]. Se also [13] or
[14] for practical criterion based on Lyapunov type criterion and we refer for
example to [26] or [37] for surveys on transportation inequality. One of the
main aspect of transportation inequality is their tensorization property, i.e.
µ⊗n will satisfy some transportation measure if µ does (with dependence on
the dimension n2/p−1) . One important development was to consider such a
propertyfordependentsequencessuchasMarkov chains. In[20],Djelloutet
al., generalizing resultof Marton [34], have provided conditions underwhich
the law of a homogeneous Markov chain (Y ) on En satisfies the Lp-
k 1≤k≤n
transportation cost-information inequality T with respect to the metric
p
n 1/p
d (x,y) := d(x ,y )p .
lp i i
!
i=1
X
We will follow similar ideas here to establish the Lp- transportation cost-
information inequality for the law of a bifurcating Markov chain (X )
i 1≤i≤N
on EN. This will allow us to obtain concentration inequalities for bifurcat-
ing Markov chains under hypotheses largely weaker than those of Bitseki
et al. [8]. It would also be tempting to generalize the approach of [28] to
Markov chains and bifurcating Markov chains to get directly deviation in-
equalities forMarkov chains, w.r.t. theinvariantmeasure. However itwould
need to restrict to reversible Markov chains and thus not directly suited to
bifurcating Markov chains and would thus recquire new ideas.
6
Remark 1.7. There are natural generalizations of the T inequality often
1
denoted α T inequality, where α is a non negative convex lower semi
1
−
continuous function vanishing at 0. We say that the probability measure µ
satisfies α T (C) if for any probability measure ν
p
−
α(W (ν,µ)) 2CH(ν/µ).
1
≤
The usual T inequality is then the case where α(t) = t2. Gozlan [23]
1
hasgeneralized Bobkov-G¨otze’s Laplacetransformcontrol [11]andDjellout-
Guillin-Wu [20] integrability criterion to this setting enabling to recover sub
or super Gaussian concentration. The result of the following section can
be generalized to this setting, however adding technical details and heavy
notations. Details will thus be left to the reader.
2 Transportation cost-information inequalities for
bifurcating Markov chains
Let (X ,i T) be a bifurcating Markov chain on E with T-probability
i
∈
transition P and initial measure ν. For p 1 and C > 0, we consider the
≥
following assumption that we shall call (H (C)) in the sequel.
p
Assumption 2.1 (H (C)).
p
(a) ν T (C);
p
∈
(b) P(x, , ) T (C), x E ;
p
· · ∈ ∀ ∈
(c) Wd(P(x, , ),P(x˜, , )) qd(x,x˜), x,x˜ E and some q > 0.
p · · · · ≤ ∀ ∈
It is important to remark that under (H (C)),(c) we have that there
p
exists r and r smaller than q such that for b = 0,1
0 1
Wd(P (x, ),P (x˜, )) r d(x,x˜), x,x˜ E.
p b · b · ≤ b ∀ ∈
Note also that when P(x,dy,dz) = P (x,dy)P (x,dz), then these last two
0 1
stability results in Wasserstein contraction implies (H (C)),(c) with q
p
(rp+rp)1/p (usingtrivialcoupling). Wemayremarkalsothatby(H (C)),(b≤),
0 1 p
P and P also satisfies (uniformly) a transportation inequality. Let us note
0 1
that thanks to the H¨older inequality, (H (C)) implies (H (C)).
p 1
We do not suppose here that q, r and r are strictly less than 1, and thus
0 1
the two marginal chains, as well as the bifurcating one, are not a priori
7
contractions. We are thus considering here both ”stable” and ”unstable”
cases.
We then have the following result for the law of the whole trajectory on
the binary tree.
Theorem 2.2. Let n ∈ N and let P be the law of (Xi)1≤i≤Tn and denote
N = T . We assume Assumption 2.1 for 1 p 2. Then T (C )
n p N
| | ≤ ≤ P ∈
where
CN2/p−1 if q < 1
(1−q)2
C = Cexp 2 2 N2/p+1 if q = 1
N  − p

 (cid:16) ex(cid:17)p(q−1)rpN 2/p
C(N +1) if q > 1.
rp−1
 (cid:16) (cid:17)
Before the proofof this result, let us make thefollowing notations. For a
Polish space χ, wedenote by (χ) thespace of probability measures on χ.
1
M
For x EN, xi := (x , ,x ). For µ (EN), let (x , ,x ) EN be
1 i 1 1 N
∈ ··· ∈M ··· ∈
distributed randomly according to µ. We denote by µi the law of x2i+1, and
by µi the conditional law of (x ,x ) given x2i−1 with the convention
x2i−1 2i 2i+1
µ1 = µ1, where x0 = x is some fixed point. In particular, if µ is the
x0 0
law of a bifurcating Markov chain with T-probability transition P, then
µi = P(x , , ).
x2i−1 i · ·
For the convenience of the readers, we recall the formula of additivity of
entropy (see for e.g. [37], Lemma 22.8).
Lemma 2.3. Let N N, let χ , ,χ be Polish spaces and , (χ)
1 N 1
where χ = N χ . T∈hen ··· P Q ∈ M
i=1 i
Q N
H( ) = H( i i ) (dx)
Q|P Qxi−1|Pxi−1 Q
i=1Zχ
X
where i and i are defined in the same way as above.
Pxi−1 Qxi−1
We can now prove the Theorem.
Proof of the Theorem 2.2. Let (EN). Assume that H( ) <
1
Q ∈ M Q|P ∞
(trivial otherwise). Let ε> 0. Theidea is of courseto doa conditionnement
with respect to the previous generation, i.e. to G but we will do it
n−1
sequentiallybypairs. Conditionallytotheirancestors,everypairofoffspring
of an individual is independent of the offspring of the other individuals for
the same generation. Let i be a member of generation G , and define for
j−1
a realization x on the tree Ti(x) := (x1,...,x|Tj|). By the definition of the
8
Wasserstein distance, there is a coupling πi of ( i , i ) such
y2i−1,x2i−1 Qy2i−1 Px2i−1
that
:= (d(y ,x )p+d(y ,x )p)dπi
Ai 2i 2i 2i+1 2i+1 y2i−1,x2i−1
Z
p
(1+ǫ)Wd i , i
≤ p Qy2i−1 Px2i−1
(cid:16) (cid:17) p
(1+ǫ) Wd i , i +Wd i , i
≤ p Qy2i−1 Py2i−1 p Py2i−1 Px2i−1
h (cid:16) (cid:17) (cid:16) (cid:17)i p
(1+ǫ) Wd i ,P (y , , ) +Wd P (y , , ),P (x , , ) ,
≤ p Qy2i−1 i · · p i · · i · ·
h (cid:16) (cid:17) (cid:16) (cid:17)i
where the second inequality is obtained thanks to the triangle inequality for
the Wd distance and the equality is a consequence of the Markov property.
p
By Assumption 2.1, and the convexity of the function x xp, we obtain,
7→
for a,b > 1 such that 1/a+1/b = 1,
p
(1+ǫ) 2CH (y2i−1)+qd(y ,x )
i i i i
A ≤
(cid:18)q (cid:19)
p
(1+ǫ) ap−1 2CH (y2i−1) +bp−1qpdp(y ,x )
i i i
≤
(cid:18) (cid:18)q (cid:19) (cid:19)
where H (y2i−1) = H( i i ). By recurrence, it leads to the finite-
i Qy2i−1|Py2i−1
ness of p-moments. Taking the average with respect to the whole law and
summing on i, we obtain
|Tn−1|
E( )
i
A
i=0
X
|Tn−1| |Tn−2|
(1+ε) ap−1(2C)p/2 E H (Y2i−1)p/2 + bp−1qp E( ) .
i i
≤    A 
Xi=1 h i Xi=0
   
Letting ε goes to 0+, we are led to
|Tn−1|
E( )
i
A
i=0
X
N |Tn−2|
ap−1(2C)p/2E H (Yi−1)p/2 + bp−1qp E( ) .
i i
≤  A 
Xi=1(cid:16) h i(cid:17) Xi=0
 
9
Iterating the latter inequality, increasing some terms and thanks to H¨older
inequality, we obtain
|Tn−1| N i N N−i
E( ) h bp−1qp N−i = h bp−1qp j
i j i
A ≤  
i=0 i=1 j=1 i=1 j=0
X X X (cid:0) (cid:1) X X(cid:0) (cid:1)
  2−p
N p/2 N N−i 2−2p 2
h2/p bp−1qp j
≤ i    
!
i=1 i=1 j=0
X X X(cid:0) (cid:1) 
 
 
whereh = ap−1(2C)p/2E[H (Yi−1)p/2]. By thedefinitionof theWasserstein
i i
distance, the additivity of entropy and using the concavity of the function
x xp/2 for p [1,2], we obtain
7→ ∈
2−p
2 2
N N−i 2−p
Wdlp( , )p ap−1(2CH( ))p/2 bp−1qp j
p  
Q P ≤ Q|P  
i=1 j=0
X X(cid:0) (cid:1) 
 
 N−1 
ap−1(2CH( ))p/2N1−p2 bp−1qp j.
≤ Q|P
j=0
X (cid:0) (cid:1)
When q < 1, we take b = q−1, so that bp−1qp = r < 1 and the desired result
follows easily. When q 1, we take b = 1+1/N and the results follow from
≥
simple analysis and this ends the proof.
Remark 2.4. For q < 1, we then have that the constant C of T inequality
N 1
for increases linearly on the dimension N. However, for T this constant
2
P
is independent of the dimension as in the i.i.d. case.
Remark 2.5. As we will see in the next section, still when q < 1, Theorem
2.2 andTheorem1.5 appliedtoF(X , ,X )= (1/N) N f(X )(where
1 ··· N i=1 i
f is a Lipschitzian function defined on E) gives us deviation inequalities
P
with a good order of N. But, when they are applied to F(X , ,X ) =
1 N
···
f(X ), deviation inequalities that we obtain does not furnish the good or-
N
der of N when N is large. The same remark holds when F(X , ,X )=
1 N
···
g(X ,X ,X ) with n 1, ,(N N[2]) and g a Lipschitzian func-
n 2n 2n+1
∈ { ··· − }
tiondefinedonE3. AsthislastquestionisimportantfortheL1-transportation
cost-information inequality of theinvariant measureof abifurcatingMarkov
chain, we give the following results.
10