Optimized Regularity Estimates of Conditional
Distribution of the Sample Mean
Victor Chulaevsky
Department of Mathematics, University of Reims Champagne-Ardenne, B.P. 1039 51687 Reims Cedex 2, France
Copyright c⃝2015 Horizon Research Publishing All rights reserved.
Abstract
We prove an optimized estimate for the reg-ularity of the conditional distribution of the empiric mean of a finite sample of IID random variables, conditional on the sample ”fluctuations”. Prior results, based on bounds in probability, provided a H¨older-type regularity of the conditional distribution. We establish a Lipschitz regularity, using bounds in expectation. The new estimate, extending a well-known property of Gaussian IID samples, is a crucial ingredient of the Multi-Scale Analysis of multi-particle Anderson-type random Hamiltonians in a Euclidean space. In particular, the H¨older regularity of the multi-particle eigenvalue distribution, sufficient for the localization analy-sis of N-particle lattice Hamiltonians, withN ≥3, needs to be replaced by Lipschitz regularity for similar Hamiltonians in the Euclidean space.Keywords
Multi-particle Anderson Localization, Eigen-value Concentration Estimates1
Introduction
1.1
The general probabilistic framework
Consider a sample ofn >1IID (independent and identi-cally distributed) random variablesX1, . . . , Xn with
Gaus-sian distribution N(0,1), and introduce the sample mean ξ=ξnand the ”fluctuations”ηirelative to the mean:
ξn= 1
n n
∑
i=1
Xi, ηi=Xi−ξn, i= 1, . . . , n.
It is well-known from standard courses of the probability the-ory (cf., e.g., [11]) that ξn is independent from the
sigma-algebraFη generated by{η1, . . . , ηn}(the latter are linearly
dependent and have rankn−1). To see this, it suffices to note that allηi are orthogonal to ξn with respect to the standard
scalar product in the linear space formed by X1, . . . , Xn,
given by
⟨Y, Z⟩:=E[Y Z],
whereY andZ are real linear combinations ofX1, . . . , Xn
(recall:E[Xi] = 0).
Therefore, the conditional probability distribution of ξn
given Fη coincides with the unconditional one, so ξn ∼
N(0, n−1), thusξ
nhas bounded density pξ(t) =
e−12t 2 √
2πn−1 ≤
n1/2
√
2π.
Moreover, for any intervalI⊂Rof length|I|, we have
ess supP{ξn(ω)∈IF
}
=P{ξn(ω)∈I} ≤
√ n
2π|I|.
In some applications to the eigenvalue analysis for random Hamiltonians, discussed below in Section 2 and in subsection 1.2, one has to estimate the probabilities of the form
P{ξn(ω)∈I(η)}=P{ξn(ω)∈[˜µ(η),µ˜(η) +s]},
(1.1) whereµ˜(·)is some measurable function ofη1, . . . , ηn, so the
intervalI(η) = [˜µ(η),µ˜(η) +s]is determined only by the fluctuations{ηi, i= 1, . . . n}. For example, withn= 2,
ξ=ξ2=
X1+X2
2 , η=η1=
X1−X2
2 ,
one may consider the probability P{ξ∈[η2, η2+s]}
= (2π)−1
∫
R2
dX1dX2e−
1 2(x
2 1+x22)1
A(x1, x2),
where the set A := {(x1, x2)∈R2
}
is defined by the in-equalities
(x1−x2)2
4 ≤
x1+x2
2 ≤
(x1−x2)2
4 +s, s >0.
In this particular case – for Gaussian samples – the con-ditional regularity of the sample mean ξn (given the
fluc-tuations) is granted, but this is not always so, as shows the following elementary example where the common prob-ability distribution of the sample X1, X2 is very regular:
Xi∼Unif([0,1]), soXiadmit a compactly supported
prob-ability density bounded by 1. In this simple example the random vector(X1, X2)is uniformly distributed in the unit
square [0,1]2, and the conditionη = c selects in the
two-dimensional plane with coordinates(X1, X2)a straight line
distribution ofξgiven{η=c}is the uniform distribution on the segment
Jc :={(x1, x2) : x1−x2= 2c,0≤x1, x2≤1}
of length vanishing at 2c = ±1. For |2c| = 1, the con-ditional distribution of ξon Jc is concentrated on a single
point, which is the ultimate form of singularity.
The above mentioned remarkable particularity of the Gaussian distributions enables one, among other things, to single out one component of a given sample and analyze its impact on one or another functional of the sample of size n > 1, while keeping fixed the remainingn−1degrees of freedom. More to the point, in the spectral analysis of ran-dom operators (cf. Section 2), the sample mean modulates a scalar operator (a multiple of the identity operator), which shifts all eigenvalues in an explicit way, while the effect of the fluctuations is much harder to control quantitatively. The question of regularity of the probability distribution of the sample meanξn with more general (in particular, singular
continuous) marginal probability measures, conditional on all the ”fluctuations”, seems to be interesting in itself; for the author of these lines, however, the motivation came from the spectral theory of random operators.
For the moment, we can address only sufficiently regu-lar marginal probability measures, which leaves the room for further research in this direction.
The main result of the present paper is given by Theorem 8.1, generalizing the estimates from Theorem 6.1 and Theo-rem 7.1.
1.2
Motivation: random
N
-particle
Hamilto-nians
The celebrated Anderson localization theory, originated in the seminal paper by P. W. Anderson [1] in 1958 (Nobel Prize in 1977) studies the evolution of quantum particles in a disor-dered media. Mathematically speaking, one has to study the decay properties of the Green functions, eigenfunctions and eigenfunction correlators of random quantum Hamiltonians (see, e.g., the monographs [8, 9, 15] for a general introduction to the Anderson localization theory). It turns out that the free propagation of particles or waves can be strongly inhibited by disorder. The crucial role, both from mathematical and phys-ical point of view, is played by the so-called ”resonances”, or ”small denominators”; the latter appear in a number of prob-lems in mathematical physics and beyond. In simple terms, one has to assess the regularity of the eigenvalue distribu-tion of random self-adjoint operators (or finite-dimensional symmetric matrices). The first fairly general estimate of such kind was proved in the celebrated paper by Wegner [16]. To-day there are various approaches leading to the Wegner-type EVC bounds (cf. [8, 15]) more general and more optimal that those relying on the results of the present paper. However, the situation is quite different in the area of multi-particle disor-dered quantum systems with a nontrivial particle inter-action. A considerable number of challenging problems here remain open due to the lack of adequate methods for assess-ing the eigenvalue concentration and eigenvalue comparison in interactive systems.
Theory of multi-particle Anderson Hamiltonians is a rel-atively new direction in this field (cf., e.g., [2, 4, 9]), and it presents greater challenges in the EVC estimation. The
main reason is that already forN = 2particles, the eigenval-ues are ”composite” quantities, and the resonances between them, in the form of ”small denominators”, are structural in nature and cannot be attributed to one or another specific lo-cus in the physical, single-particle configuration space. As a result, correlations between composite EVs do not decay with distance; these strong correlations become particularly difficult to deal with starting fromN = 3. Suffices to say that there still are open problems in the area ofN-particle Ander-son Hamiltonians in a Euclidean space, while the situation is somewhat simpler for the lattice Hamiltonians (cf. [10]).
A partial solution to this general problem, initially pro-posed in [6], used the specific property of Gaussian distri-butions which we discussed in subsection 1.1. In a modi-fied form, it was extended to the IID random potentials with uniform marginal distributions and ”smooth” perturbations thereof; see [5, 10]. However, the H¨older regularity (of any orderθ < 1) of theN-particle EV distribution used in the context of lattice Hamiltonians turns out to be insufficient for the applications to the N-particle Anderson models in Rd,
where one needs the Lipschitz continuity of the EV distribu-tion, and this is the main subject of the present paper.
We hope that the approach used in [6] and in this paper may prove useful in other areas of mathematical physics and applications of the probability theory.
A detailed discussion of the interactive Anderson models, especially in the Euclidean space, is certainly well beyond the scope of this short note; an introduction to this relatively recent direction of research in mathematical physics of ran-dom operators can be found in the monograph [9]. However, it is to be emphasized that the results of this paper are crucial for the rigorous proof of uniformly strong decay bounds on eigenfunction correlators with respect to the natural distance in the multi-particle configuration space and not in the so-called Hausdorff distance, used in the works [7, 13, 12] and some other papers. We also stress that the new EVC bounds are required for the proof of localization in a physically re-alistic geometrical setting (in finite-size disordered samples); see the discussion in [10].
2
An application to the Wegner-type
bounds
LetΛbe a finite graph, with|Λ| =n ≥1, andH(ω) =
HΛ(ω)be a discrete Schr¨odinger operator (cf., e.g., [8, 9]) acting in the finite-dimensional Hilbert space H = HΛ =
ℓ2(Λ), with an IID random potentialV : Λ×Ω→R, relative
to some probability space(Ω,F,P). For example, one can takeΛ = [−L, L]∩Z1and define the random, second order
discrete Schr¨odinger operatorH(ω) :ℓ2(Λ)→ℓ2(Λ)by (
H(ω)ψ)(x) =((H0+V(ω))ψ
)
(x)
:=−ψ(x−1)−ψ(x+ 1) +V(x, ω)ψ(x),
with Dirichlet boundary conditions (cf. [8, 9]) outside
[−L, L]; hereV : Λ×Ω→Ris a random field onΛrelative to some probability space(Ω,B,P). On an arbitrary con-nected finite graph Λ, one usually takes the kinetic energy operatorH0given by the negative graph Laplacian (cf., e.g.,
[3]), (
(−∆Λ)ψ
)
(x) = ∑
y∈Λ: d(x,y)=1
(
hered(x, y)is the canonical graph-distance onΛ, defined as the shortest path fromxtoy over the edges of the graph in question.
Decomposing the random fieldV onΛ, V(x;ω) =ξn(ω) +ηx(ω),
we can representH(ω)as follows: H(ω) =ξn(ω)1+A(ω),
where the self-adjoint operatorA(ω)
A=−∆Λ+ηx(ω),
(hereηxis identified with the operator of multiplication by
the ”residual”, ”fluctuation” random potentialx 7→ ηx(ω))
is Fη-measurable, and so are its eigenvalues µj(ω), j = 1, . . . , n. Since A(ω) commutes with the scalar operator ξn(ω)1, the eigenvaluesλj(ω)ofH(ω)have the form
λj(ω) =ξn(ω) +µj(ω). (2.1)
The numeration of the eigenvaluesλj(ω),µj(ω)is of course
not canonical, but they can be consistently defined as random variables onΩ.
The representation (2.1) implies immediately the follow-ing EVC bound. Fix and interval Is = [t, t+s] and let PIs(H(ω)be the spectral projection of operatorH(ω)to the intervalIs(cf., e.g., [14]), then
P{trPIs(H(ω))≥1} ≤
n
∑
j=1
P{λj(ω)∈Is}
= n
∑
j=1
E[P{ξn+µj∈IsFη
} ]
= n
∑
j=1
E[P{ξn ∈[−µj+t,−µj+t+s]Fη
} ] (2.2)
Further, omitting the argument ω for brevity and setting
˜
µj(ω) :=−µj(ω) +t, we have
P{ξn+µj∈IsFη
}
=P{ξn∈[−µj+t,−µj+t+s]Fη
}
=P{ξn∈[˜µj,µ˜j+s]Fη
} (2.3)
where µ˜j(ω) are Fη-measurable, i.e., fixed under the
conditioning. Therefore, the unconditional probability P{λj ∈Is}can be assessed by analyzing the regularity of
the conditional probability distribution ofξn figuring in the
RHS of (2.3).
3
Reduction to the local analysis in the
sample space
Assume that the supportS ⊂Rof the common continu-ous marginal probability measure of the IID random variables Xj,1≤j ≤n, is covered by a finite or countable union of
intervals:
S ⊂ ∪k∈KJk, K ⊂Z, Jk= [ak, bk], ak+1 ≥bk.
LetK=Kn, and for eachk= (k1, . . . , kn)∈K, denote
Jk= n
×
i=1
Jki.
Owing to the continuity of the marginal measure,Jk are
es-sentially disjoint: two distinct intervalsJk, Jlcan only have
common endpoints. Respectively, the family of the paral-lelepipeds{Jk, k∈ K}forms a partitionKof the sample space, which we will often identify with the probability space
Ω. Further, letFK be the sub-sigma-algebra ofFgenerated by the partitionK. The probabilities of the general form (1.1) can be assessed as follows:
P{ξn∈[˜µ,µ˜+s]}
=E[P{ξn ∈[˜µ,µ˜+s]FK
} ]
= ∑
k∈K
P{Jk}P{ξn ∈[˜µ,µ˜+s]Jk
} .
LetPk{·}be the conditional probability measure, given the event{X∈Jk},Ek[·]the respective expectation, andpk= P{Jk}. Then we have
P{ξn∈[˜µ,µ˜+s]}
= ∑
k∈K
pkEk[Pk{ξn∈[˜µ,µ˜+s]Fη
} ]
≤sup
k∈KEk [
Pk{ξn∈[˜µ,µ˜+s]Fη
} ] .
(3.1)
Therefore, one may seek a satisfactory bound on the LHS of (3.1) by assessing the ”local” conditional probabilities Pk{ξn∈[˜µ,µ˜+s]Fη
}
, where each random variableXj
is restricted to a subintervalJkj of its global support, so the entire sampleX = (X1, . . . , Xn)is restricted to the
paral-lelepipedJk⊂Rn.
In the next section, we perform such analysis in the case of a uniform distribution of the IID variablesXi.
4
Uniform
marginal
distributions.
General setup
Let be given a real numberℓ > 0 and an integern ≥ 2. Consider a sample of n IID random variables with uni-form distributionUnif([0, ℓ]), and introduce again the sample meanξ=ξnand the ”fluctuations”ηirelative to the mean:
ξn = 1
n n
∑
i=1
Xi, ηi=Xi−ξn.
Further, consider then-dimensional Euclidean space of real linear combinations of the random variablesXi. Clearly, the
variablesηi:Rn→Rare invariant under the group of
trans-lations
(X1, . . . , Xn)7→(X1+t, . . . , Xn+t), t∈R,
and so are their differencesηi−ηj ≡Xi−Xj,1≤i < j≤ n. Introduce the variables
Yi=ηi−ηn, 1≤i≤n−1. (4.1)
Then the spaceRnis stratified into a union of affine lines of the form
L(Y) :={X ∈Rn : ηi−ηn=Yi, i≤n−1} ={X∈Rn : Xi−Xn =Yi, i≤n−1},
(4.2)
labeled by the elementsY = (Y1, . . . , Yn−1)of the(n−1)
(1, . . . ,1). Denote
X(Y) =L(Y)∩[0, ℓ]n
={X∈[0, ℓ]n: Xi−Xn =Yi, i≤n−1}
and endow each nonempty interval X(Y) ⊂ Rn with the natural structure of a probability space inherited fromRn:
• if|X(Y)| = 0(an interval reduced to a single point), then we introduce the trivial sigma-algebra and the counting measure;
• if |X(Y)| = r > 0, then we use the inherited struc-ture of an interval of a one-dimensional affine line and the normalized measure with constant densityr−1with
respect to the inherited Lebesgue measure onX(Y). The transformation X 7→ (ξn, η1, . . . , ηn−1) is
non-degenerate, but not orthogonal. We will have to work with the metric onX(Y), induced by the standard Riemannian metric in the ambient spaceRn; to this end, introduce an orthogonal
coordinate transformation inRn,X 7→ ( ˜ξ
n,η˜1, . . . ,η˜n−1),
such that
˜
ξn=n−1/2 n
∑
i=1
Xi=n1/2ξn; (4.3)
the exact form ofη˜j,j= 1, . . . , n−1is of no importance for
our analysis, provided that the transformation is orthogonal.
Remark4.1. For later use, note that, due to (4.3), each of the re-scaled variablesn1/2Xican serve as the normalized length
parameter on the elementsX(Y). Along an elementX(Y), one can simultaneously parameterizeξ˜and the variablesXi,
by settingξ˜(t) =c0+t,Xj(t) =cj+n−1/2t, with arbitrarily
chosen constantscj. Here,ξ˜n is a natural length parameter
onX(Y), since the transformationX 7→( ˜ξn,η˜1, . . . ,η˜n−1)
is orthogonal.
5
Short intervals are unlikely
Lemma 5.1. Assume that the IID random variables
X1, . . . , Xn,n ≥ 2, are uniformly distributed in an
inter-val[0, ℓ],ℓ >0. Then for allt∈(0, ℓ/2]one has
P{ |X(Y)|< t} ≤t2n/ℓ2. (5.1)
Proof. Let
X =X(X) = min
i Xi, X =X(X) = maxi Xi. (5.2)
WhileX(X)andX(X)vary along the elementsX(Y), their differenceX(X)−X(X)does not; it is uniquely determined byX(Y).
According to Remark 4.1, eachn1/2Xi,i = 1, . . . , n,
re-stricted to X(Y), provides a normalized length parameter onX(Y); thus the range of eachn1/2X
i|X(Y) is an
inter-val of length |X(Y)|. One can increase (resp., decrease), e.g., the value of X1, as long as all {Xi,1 ≤ i ≤ n}
are strictly smaller than ℓ (resp., strictly positive). There-fore, the maximum increment of X1 (indeed, of any Xi)
alongX(Y)is given byℓ−X(X), and its maximum decre-ment equals X(X), so the range of the normalized length parametern1/2X
1 along X(Y(X))is an interval of length
n1/2(ℓ−X(X) +X(X)):
|X(Y(X))|=n1/2(ℓ−X(X) +X(X)). (5.3)
Since both X(X)andℓ−X(X)are non-negative, one has the implication
|X(Y)|=X+ (ℓ−X)< t =⇒ max{X, ℓ−X}< t.
With 0 ≤ t ≤ ℓ/2,(ℓ−Xi < t
)
implies(Xi > t
) , thus denoting
Aij(t) :={Xi< t} ∩ {ℓ−Xj< t}, (5.4)
we have, for anyi,
Aii(t) ={Xi < t} ∩ {ℓ−Xi< t}=∅. (5.5)
Therefore, {
max{X, ℓ−X}< t }
⊂∪
i̸=j
{
Xi< t, ℓ−Xj< t
} .
Thus the union∪i̸=jAij(t)contains all the samplesX with
|X(Y)|< t.
The sample{Xk}is IID, with common probability density
bounded byρ=ℓ−1, so for anyi̸=j
P{Aij(t)}=P{Xi< t} ·P{ℓ−Xj< t}=ρ2t2.
Therefore,
P{ |X(Y)|< r}
=P
{
n1/2((ℓ−X(X)) +X(X))< r }
=P{ ((ℓ−X(X)) +X(X))< rn−1/2 }
≤∑
i̸=j
P{Aij
(
rn−1/2) }≤n(n−1)
(
ρrn−1/2 )2
≤r2n/ℓ2.
(5.6) This completes the proof.
By a change of variable, one can extend the above re-sult tonindependent random variables uniformly distributed in their individual intervalsJi = [ai, ai +ℓ], for arbitrary a1, . . . , an∈R.
6
Regularity bound for the uniform
distribution
Theorem 6.1. Let be given IID random variables X1,
. . . , Xn and a measurable functionλ : Y 7→ λ(Y). In
each intervalX(Y)⊂ L(Y)∼=R, introduce the sub-interval
Is(Y) = [λ(Y), λ(Y) +s]∩ L(Y). Then for anys∈(0,1],
one has
P{ξ(ω)∈Is(Y)} ≤3n3ℓ−1s. (6.1)
Proof. Recall that the natural length parameter on the lines L(Y)is given by ξ˜ = √nξ and not by ξ itself, since the gradient ofξis given by the vector(n−1, . . . , n−1)of norm 1/√n. Therefore, whenξruns throughIs(Y),ξ˜runs through
the intervalI˜s= [λe(Y),λe(Y) +
√
Setl(ω) := |X(Y)|, and denote by1l≥s√n the indicator function of the event{l≥s√n}, then we have
P{ξ∈Is(η)}=E
[
P{ξ˜∈I˜s(η)Fη} ]
=E
[
1l<s√nP {
˜
ξ∈I˜s(η)Fη
} ]
+E
[
1l≥s√nP
{
˜
ξ∈I˜s(η)Fη
} ]
≤P{l< s√n}+E
[
1l≥s√nP
{
˜
ξ∈I˜s(η)Fη
} ] , (6.2) where, by virtue of (5.6),
P{l(ω)< s√n}≤n(s√n)2/ℓ2=n2s2/ℓ2, (6.3) yielding
sup s>0
P{l(ω)< s√n}
s2 ≤n
2/ℓ2. (6.4)
The second summand in the last RHS in (6.2) can be assessed as follows:
E[1l≥s√nP {
˜
ξ∈I˜s(η)Fη
} ]
≤E[1l≥s√ns
l
]
=sE [
1l≥s√n(l(ω))−1
]
=s√n ∫ ℓ√n
s√n
r−1dFl(r).
(6.5) Now we apply the well-known formula of integration by parts for the Stiltjes integral (cf., e.g., [11, Sect. V.6]) giving the moment of orderαof a probability distribution with proba-bility distribution function (PDF)F(·):
∫ ∞
0
rαdF(r) =α ∫ ∞
0
rα−1F(r)dr,
valid under the assumption of convergence of the RHS in-tegral. In our case, we only have to integrate over a finite interval[s√n, ℓ√n], thus avoiding singularity atr= 0. For the PDFFl(·), using the upper bound (6.4), we obtain
∫ ℓ√n
s√n
dFl(r)
r =
F(r)
r ℓ
s√n +
∫ ℓ√n
s√n
r−2Fl(r)dr
≤ 1 ℓ√n+ℓ
√ nsup
r>0
Fl(r)
r2 ≤ 1
ℓ√n+
ℓ√n·n2
ℓ2
≤2n5/2 ℓ .
(6.6) Collecting (6.3), (6.5), (6.6), and usings/ℓ≤1, the assertion follows:
P{ξ∈Is(η)} ≤n2 s2
ℓ2 +
2n5/2·s√n
ℓ ≤
3n3
ℓ s. (6.7)
7
Smooth positive densities
Now we consider a richer class of probability distributions. While the conditions which we assume are certainly very re-strictive, they are sufficient for the applications to some phys-ically relevant Anderson models.
Theorem 7.1. Assume that the common probability distribu-tion of the IID random variablesVj, j = 1, . . . , n, with the
cumulative probability distribution functionFV, satisfies the
following conditions:
(i) the probability distribution is absolutely continuous:
dFV(v) =ρ(v)dv, suppρ= [a, a+ℓ]; (7.1)
(ii) the probability density ρ(·) has bounded logarithmic derivative on(a, a+ℓ):
(lnρ)′1(a,a+ℓ)∞≤Cρ′ <+∞. (7.2)
Then there exists a constantC =C(FV, ℓ) <∞such that
for anys∈(0, ℓn−1)and anyF
η-measurable random
vari-ableλ, settingIs(ω) := [λ(ω), λ(ω) +s], one has the
fol-lowing bound:
P{ξn(ω)∈Is(ω)} ≤Cn2s. (7.3)
Proof. Without loss of generality, it suffices to prove the claim forsuppρ= [0, ℓ], which we assume below.
As in Section 3, introduce a partition of the sample space into the cubes Jk, induced by the decomposition [0, ℓ] =
⊔kJk:
Jk= n
×
i=1Jki, k= (k1, . . . , kn), whereJk = [ak, ak+1], ak := k−1
n , k= 1, . . . , n. The hypothesis (7.2) implies that for anyx ∈ Jk the loga-rithm ofp(x)is well-defined and satisfies the upper bound
|lnp(x)−lnp(ak)| ≤ n
∑
i=1
|lnρ(xi)−lnρ(aki)| ≤ n Cp′ℓ
n ,
thus
∀x∈JK p(x) =p(ak)eαn(x),
where |αn(x)| ≤ Cℓ; with ℓ fixed in the condition (7.1),
p(x)is therefore uniformly bounded. Now introduce inJk:
• the uniform probability distribution Pk, i.e., the nor-e malized measure with constant density epk w.r.t. the Lebesgue measure;
• the probability distribution induced by P, conditional on{X∈Jk}, i.e., the normalized measure with density
pk(x) =Zk−1p(x) =
p(x)
∫
JkP(y)dy
.
Due to continuity of the densityp, we have∫J
kP(y)dy=
c|Jk|, for somec∈[e−αn,e+αn], so pk(x)
e p(x) =
p(x)
c ∈
[
e−2αn,e+2αn] Hence for any eventA, we have
e−2αnP{ A } ≤Pk{A} ≤e2αnP{ A }. (7.4) Finally, it follows from (7.4) and (3.1) that
P{ξ∈Is(η)} ≤sup
k
Pk{ξ∈Is(η)} ≤C(FV, ℓ)n2s.
We will call the property of the common probability distri-bution of the IID random variablesX1, . . . , Xn, expressed by
8
Extension to convolution measures
The results of Section 7 can be easily adapted to a class of convolution probability measuresP = P1∗P2where at
least one of the measuresP1,P2satisfies the hypotheses of
Theorem 7.1.
Theorem 8.1. Assume that the common probability distribu-tionν of the IID random variablesVj, j = 1, . . . , n, admits
the representation
ν =ν1∗ν2, (8.1)
whereν1satisfies the hypotheses of Theorem 7.1. Then there
exists a constantC =C(FV, ℓ)<∞such that for anys∈ (0, ℓn−2)and anyF
η-measurable random variableλ, setting Is(ω) := [λ(ω), λ(ω) +s], one has the following bound:
P{ξn(ω)∈Is(ω)} ≤Cn3s. (8.2)
Proof. It follows immediately from the assumption (8.1) that the random variablesVjcan be represented as sumsVj(ω) = Aj(ω) +Bj(ω), where
•the family of random variables{Aj, 1 ≤ j ≤ n}is IID,
with common probability distributionν1;
•the family of random variables{Bj),1 ≤ j ≤n}is also
IID, with common probability distributionν2;
•the families{Aj}and{Bj}are mutually independent.
LetP,P1andP2be the product probability measures in the
sample space∼=Rn, generated respectively by the marginal measuresν,ν1andν2. Further, letFBbe the sigma-algebra
generated by the family{Bj, j= 1, . . . , n}. Then the
condi-tional measureP{ · |FB}is equivalent toP1, thus
P{ξn(ω)∈Is(ω)}=E(P)
[
P{ξn(ω)∈Is(ω)FB
} ]
=E(P)
[
P1{ξn(ω)∈Is(ω)}
]
≤Cn3s.
Example 1.The so-called triangular distributionν∗ν, where νis the uniform distribution in[0,1]. More generally, one can take a convolution powerν∗n,n≥ 2. This example shows
that the property SRCMcan hold in a class of probability measures with density vanishing at the edges of its support.
Example 2. The convolution ν = ν1∗ν2 of the uniform
distributionν1 = Unif([0,1])with the exponential
distribu-tionν2(with density1[0,+∞)e−t). Here the propertySRCM
holds true for upper-unbounded random variables.
As pointed out in the Introduction (cf. also [6]), the Gaussian distribution features a particularly strong form of the property SRCM. Theorem 8.1 extends SRCM to a much larger class of unbounded random variables. In par-ticular, note that the decay rate of the tail probabilities P{ |V(ω)|> t}, ast → +∞, can be quite slow. For ex-ample, one can take asν2 any stable law (cf., e.g., [11]) of
indexα ∈ [1,2]; here α = 2corresponds to the Gaussian distribution andα= 1to the Cauchy distribution.
9
Conclusion
We have shown that the well-known and widely used prop-erty of the Gaussian IID samples of random variables, viz.
the independence of the sample mean of the ”sample fluctua-tions”, has a direct analog for a much larger class of marginal probability distributions. For the moment, our approach re-quires the marginal distribution to have a smooth probability density with respect to the Lebesgue measure. Compared to prior results (cf. [5, 10]), where the conditional distribution of the sample mean (given the sigma-algebra generated by all sample fluctuations) was proved to be H¨older-continuous of orderθ= 2/3, the present paper gives a sharp H¨older expo-nentθ = 1(thus proving Lipschitz continuity of the condi-tional measure at hand).
The new result is an important component in the spectral analysis of random N-particle quantum Hamiltonians, de-scribing the quantum transport of interacting particles (e.g., electrons) in a disordered environment. Quantum transport and/or localization in interacting disordered systems is a rel-atively new direction both in theoretical and mathematical physics, where there still are many challenging open prob-lems, and our results shed a new light in this direction. They will constitute a major component in the localization analysis ofN-particle Anderson Hamiltonians with nontrivial interac-tion in a Euclidean space, in a forthcoming paper extending the results of our recent work [10] to continuous systems.
We hope that the probabilistic problem investigated in this paper, as well as the method used here, can prove useful in a more general mathematical framework where analytic as-pects intertwine with the probabilistic ones.
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