Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails

www.imstat.org/aihp 2012, Vol. 48, No. 4, 1103–1136

DOI:10.1214/11-AIHP441

© Association des Publications de l’Institut Henri Poincaré, 2012

Tail and moment estimates for chaoses generated by symmetric random variables with logarithmically concave tails ¹

Radosław Adamczak

and Rafał Latała

aInstitute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland. E-mail:R.Adamczak@mimuw.edu.pl bInstitute of Mathematics, Polish Academy of Sciences, ´Sniadeckich 8, 00-956 Warszawa, Poland. E-mail:R.Latala@mimuw.edu.pl

Received 9 July 2010; revised 21 April 2011; accepted 10 June 2011

Abstract. We present two-sided estimates of moments and tails of polynomial chaoses of order at most three generated by in- dependent symmetric random variables with log-concave tails as well as for chaoses of arbitrary order generated by independent symmetric exponential variables. The estimates involve only deterministic quantities and are optimal up to constants depending only on the order of the chaos variable.

Résumé. Nous établissons un encadrement des moments et des queues d’un chaos polynomial d’ordre au plus trois engendré par des variables aléatoires indépendantes symétriques à queues log-concaves et pour des chaos d’ordre quelconque engendrés par des variables aléatoires indépendantes symétriques exponentielles. Ces estimations ne font intervenir que des quantités déterministes et sont optimales à des constantes près qui ne dépendent que de l’ordre du chaos.

MSC:Primary 60E15; secondary 60G15

Keywords:Polynomial chaoses; Tail and moment estimates; Metric entropy

1. Introduction

A (homogeneous) polynomial chaos of orderdis a random variable defined as n

i1,...,id=1

a_i1,...,idX_i1· · ·X_id, (1)

whereX₁, . . . , X_n is a sequence of independent real random variables and(a_i1,...,id)_{1≤i1,...,id≤n}is ad-indexed sym- metric array of real numbers, satisfyinga_i1,...,id=0 whenever there existsk=lsuch thati_k=i_l.

Random variables of this type appear in many branches of modern probability, e.g. as approximations of multiple stochastic integrals, elements of Fourier expansions in harmonic analysis on the discrete cube (when the underlying variablesX_i’s are independent Rademachers), in subgraph counting problems for random graphs (in this caseX_i’s are zero–one random variables) or in statistical physics.

Chaoses of order one are just linear combinations of independent random variables and their behavior is well- understood. Chaoses of higher orders behave in a more complex way as the summands in (1) are no longer inde- pendent. Nevertheless, due to their simple algebraic structure, many counterparts of classical results for sums of independent random variables are available. Among well known results there are Khinchine type inequalities and tail

1Research partially supported by MNiSW Grant N N201 397437 and the Foundation for Polish Science.

bounds involving the variance (see e.g. [4,11,19] or Chapter 3 of [7]). There are also two-sided inequalities in terms of expectations of suprema of empirical processes, which themselves are in general difficult to estimate (see [3,5] for Gaussian chaoses and [1,18] for the general case of chaoses generated by variables with log-concave tails).

In several cases, under additional assumptions on the distribution ofX_i’s, even more precise results are known, which give two sided estimates on moments of polynomial chaoses in terms of deterministic quantities involving only the coefficientsa_i1,...,id (the estimates are accurate up to a constant depending only ond). Examples include Gaussian chaoses of arbitrary order [15], chaoses generated by nonnegative random variables with log-concave tails [16] and chaoses of order at most two, generated by symmetric random variables with log-concave tails ([10] ford=1 and [14] ford=2).

The aim of this paper is to provide some extensions of these results. In particular we provide two-sided estimates for moments of chaoses of order three generated by symmetric random variables with log-concave tails (Theorems 3.1and3.2) and for chaoses of arbitrary order, generated by symmetric exponential variables (Theorem3.4).

Before we formulate precisely our main results let us recall the notion of decoupled chaos and decoupling inequal- ities. A decoupled chaos of orderd is a random variable of the form

n i₁,...,i_d=1

a_i1,...,idX¹_i

1· · ·X^d_i

d, (2)

where(a_i1,...,id)_{1≤i1,...,id≤n} is ad-indexed array of real numbers andX_i^l,i=1, . . . , n,l=1, . . . , d, are independent random variables.

One can easily see that each decoupled chaos can be represented in the form (1) with a modified matrix and for suitably largern. However it turns out that for the purpose of estimating tails or moments of chaoses it is enough to con- sider decoupled chaoses. More precisely, we have the following important result due to de la Peña and Montgomery- Smith [8].

Theorem 1.1. Let(a_i1,...,id)_{1≤i1,...,id≤n}be a symmetricd-indexed array such thata_i1,...,id =0whenever there exists k=lsuch thati_k=i_l.LetX₁, . . . , X_nbe independent random variables and(X_i^j)_1≤i≤n,j=1, . . . , d,be independent copies of the sequence(X_i)_1≤i≤n.Then for allt≥0,

L⁻_d¹P

n i₁,...,i_d=1

a_i1,...,idX¹_i

1· · ·X_i^d

d

≥L_dt

≤P

n i₁,...,i_d=1

a_i1,...,idX_i1· · ·X_id ≥t

≤L_dP

n i₁,...,i_d=1

a_i1,...,idX¹_i

1· · ·X^d_i

d

≥L⁻_d¹t

,

whereL_d∈(0,∞)depends only ond.In particular,for allp≥1, L˜⁻_d¹

n i1,...,id=1

ai₁,...,i_dX_i¹

1· · ·X_i^d

d

p

≤

n i1,...,id=1

ai₁,...,i_dXi₁· · ·Xi_d

p

≤ ˜Ld

n i1,...,id=1

ai₁,...,i_dX¹_i

1· · ·X^d_i

d

p

,

whereL˜_ddepends only ond.

If we are not interested in the values of numerical constants, the above theorem reduces estimation of tails and moments of general chaoses of orderd to decoupled chaoses. The importance of this result stems from the fact that the latter can be treated conditionally as chaoses of smaller order, which allows for induction with respect to d.

Since the reduction is straightforward, in the sequel when formulating our results we will restrict our attention to the decoupled case.

Let us finish the introduction by remarking that two-sided bounds on moments of chaoses of the form (1) can be used to give two-sided estimates for more general random variables, i.e. tetrahedral polynomials inX₁, . . . , X_d, e.g. to polynomials in which every variable appears in a power at most 1. This is thanks to the following simple observation, which to our best knowledge has remained unnoticed.

Proposition 1.2. For j =0,1, . . . , d let (a^j_i

1,...,ij)_{1≤i1,...,ij≤n} be aj-indexed symmetric array of real numbers(or more generally elements of some normed space),such thata_i^j

1,...,i_j =0if i_k=i_l for some1≤k < l≤j (forj =0 we have just a single numbera_∅⁰).LetX1, . . . , Xnbe independent mean zero random variables.Then there exists a constantLd∈(0,∞),depending only ond,such that for allp≥1,

d j=0

n i1,...,ij=1

a_i^j

1,...,i_jX_i1· · ·X_ij p

≤L_d

d j=0

n i1,...,ij=1

a_i^j

1,...,i_jX_i1· · ·X_ij p

.

Note that a reverse inequality boils down just to the triangle inequality inL_pand so the above proposition imme- diately gives two-sided estimates of moments of tetrahedral polynomials from estimates for homogeneous chaoses.

Since the details are straightforward we will not state explicitly the results which can be obtained from the inequal- ities we present. The easy (given general results on decoupling) but notationally involved proof of Proposition1.2is deferred to theAppendix.

The organization of the article is as follows. After introducing the necessary notation (Section2) we state our main results (Section3) and devote the rest of the paper to their quite involved proof. In the course of the proof we provide entropy estimates for special kinds of metrics on subsets of certain product sets (Section5.2) as well as bounds on empirical processes indexed by such sets (Sections 6 and7 where we also provide some partition theorems). We believe that these results may be of independent interest. In Section8we conclude the proof of our result for chaoses of order three and in Section9we give a proof of estimates for chaoses of arbitrary order generated by exponential variables.

2. Definitions and notation

Let(X_i^j)_{1≤i≤n,1≤j≤d}be a matrix of independent symmetric random variables with logarithmically concave tails, i.e.

such that the functionsN_i^j:[0,∞)→ [0,∞]defined by N_i^j(t)= −lnPX_i^j≥t

are convex. We assume that r.v.’s are normalized in such a way that inf t≥0:N_i^j(t)≥1

=1. (3)

We set Nˆ_i^j(t)=

t² for|t| ≤1, N_i^j

|t|

for|t|>1.

Remark. When working withd=1we will suppress the upper indexj and write simplyX_i orN_i. Recall that thepth moment of a real random variableXis defined asX^pp=E|X|^p.

For a sequence (xi)i of real numbers (sometimes multiindexed) we will denote x2=

ix_i² and x1=

i|x_i|.

Fori∈ {1, . . . , n}^d andI⊂ {1, . . . , d}we writeiI=(i_k)_k∈I. ByP_d we will denote the family of all partitions of {1, . . . , d}into nonempty, pairwise disjoint subsets. ForJ = {I1, . . . , Ik} ∈Pd,p≥2 and a multiindexed matrix(ai) we define

(a_i)^N

J,p=

s1∈I1,...,sk∈Ik

sup

i

a_i k l=1

x_i^l

Il:

i_sl

Nˆ_i^sl

slx_i^l

Il

i_Il\{sl}

2

≤p,1≤l≤k

. (4)

Remark. WhenI_lis a singleton,i.e.I_l= {s_l},then for any fixed value of the indexi_sl,(x_i^l

Il)_i

Il\{sl}2= |x_i^l

sl|.

In particular ford=3 we have (aij k)^N

{1,2,3},p=sup

ij k

aij kxij k:

i

Nˆ_i¹

j,k

x²_{ij k}

≤p

+sup

ij k

a_{ij k}x_{ij k}:

j

Nˆ_j²

i,k

x_{ij k}²

≤p

+sup

ij k

a_{ij k}x_{ij k}:

k

Nˆ_k³

i,j

x_{ij k}²

≤p

,

(a_{ij k})^N

{1,2}{3},p=sup

ij k

a_{ij k}x_ijy_k:

i

Nˆ_i¹

j

x_ij²

≤p,

k

Nˆ_k³(y_k)≤p

+sup

ij k

a_{ij k}x_ijy_k:

j

Nˆ_j²

i

x_ij²

≤p,

k

Nˆ_k³(y_k)≤p

and

(a_{ij k})^N_{

1}{2}{3},p=sup

ij k

a_{ij k}x_iy_jz_k:

i

Nˆ_i¹(x_i)≤p,

j

Nˆ_j²(y_j)≤p,

k

Nˆ_k³(z_k)≤p

.

ByB₁ⁿandB₂ⁿwe will denote respectively the standardⁿ₁ andⁿ₂ balls, e.g.B₁ⁿ= {x∈Rⁿ: x1≤1}andB₂ⁿ= {x∈Rⁿ: x2≤1}.

Throughout the article we will writeLd, Lto denote constants depending only ond and universal constants re- spectively. In all cases the values of a constant may differ at each occurence.

ByA∼dBwe mean that there exists a constantL_d∈(0,∞), such thatL⁻_d¹B≤A≤L_dB. We will also denoteX^j=(X_i^j)_1≤i≤nand writeEjfor the expectation with respect toX^j.

3. Main results

Theorem 3.1. For anyd≥1andp≥2we have

i

a_iX¹_i

1· · ·X_i^d

d

p

≥ 1 L_d

J∈P_d

(a_i)^N

J,p. (5)

Theorem 3.2. Ford≤3andp≥2,

i

a_iX¹_i

1· · ·X_i^d

d

p

≤L_d

J∈Pd

(a_i)^N

J,p. (6)

Remark 1. LetX_i^j=cg_i^j,whereg_i^j are i.i.d.N(0,1)r.v.’s and1< c <10/9is a constant for which the normaliza- tion(3)holds.Thent²/L≤ ˆN_i^j(t)≤Lt²and forJ = {I₁, . . . , I_k} ∈P_d,p≥2

(ai)^N

J,p∼dp^k/2(ai)

J, where

(ai)

J =sup

i

ai

k l=1

x_i^l

Il: x_i^l

Il

2≤1,1≤l≤k

. (7)

Theorems3.1and3.2(for arbitraryd)in this case were established in[15].

A standard application of the Paley–Zygmund inequality (see e.g. Corollary 3.3.2 of [7]) and the fact thatpth and 2pth moments of chaoses generated by random variables with log-concave tails are comparable up to constants depending only on the order of the chaos yield the following corollary (for details see the proof of Corollary 1 in [15]).

Corollary 3.3. Ford≤3andt >0, 1

L_de^{−t /Ld} ≤P

i

aiX_i¹

1· · ·X^d_i

d

≥

J∈P_d

(ai)^N

J,t

≤Lde^{−t Ld}.

We are not able to show Theorem3.2ford >3 in the general case. However we know that it holds for exponential random variables.

Theorem 3.4. IfN_i^j(t)=tfor alli, j andt >0,then for anyd≥1andp≥2the estimate(6)holds.

4. Proof of Theorem3.1

We will proceed by induction with respect tod. The cased=1 was proved in [10]. Let us therefore assume that the theorem holds for all positive integers smaller thand >1.

Note that since we allow the constants to depend ond, it is enough to show that the left-hand side of (5) is minorized by each of the summands on the right-hand side.

For anyJ = {I1, . . . , Ik} ∈Pd, withk≥2, the induction assumption applied conditionally on(X^j)j∈I₁ gives

i

a_iX_i¹

1· · ·X^d_i

d

p

≥ 1 L_d−#I1

EI₁

iI1

a_i

r∈I1

X^r_i

r

i_{I c}

1

^N

J\{I₁},p

p1/p

, (8)

whereN=(N_i^j)_{1≤i≤n,j∈I}^c

1.

Let us fix arbitrarys1∈I1, . . . , sk∈Ik. We have EI₁

iI1

a_i

r∈I₁

X^r_i

r

i_{I c}

1

^N

J\{I1},p

p

≥EI1

sup

i_{I c}

1

i_I1

a_i

r∈I1

X_i^r

r

k

l=2

x_i^l

Il

:

i_sl

Nˆ_i^l

slx_i^l

Il

i_Il\{sl}

2

≤p,2≤l≤k p

≥sup

EI₁

i_{I c}

1

iI1

ai

r∈I1

X^r_i

r

k

l=2

x_i^l

Il

p

:

i_sl

Nˆ_i^l

slx_i^l

Il

i_Il\{sl}

2

≤p,2≤l≤k

≥ 1 L^p_#I

1

sup

i

a_i k l=1

x_i^l

Il

:

i_sl

Nˆ_i^l

slx_i^l

Il

i_Il\{sl}

2

≤p,1≤l≤k p

,

where the last inequality follows from another application of the induction assumption, this time to a chaos of order

#I1. Since the indicess₁, . . . , s_d run over sets of cardinality not exceeding d, the above estimate together with (8) imply that

i

aiX_i¹

1· · ·X^d_i

d

p

≥ 1

L_dai^N_J,p.

The casek=1 requires a different approach. Again it is enough to show that for eachl∈ {1, . . . , d},

i

a_iX¹_i

1· · ·X_i^d

d

p

≥ 1

L_dsup

i

a_ix_i:

i_l

Nˆ_i^l

l

i_{l}c

x_i² 1/2

≤p

.

Consider anyx_isuch that

i_lNˆ_i^l

l((

i_{l}cx_i²)^1/2)≤p. By the symmetry ofX^l_i we have

i

a_iX¹_i

1· · ·X_i^d

d

^p

p

=ElE{l}^c

i_l

X^l_i

l

i_{l}c

a_i

k=l

X^k_i

k

^p

≥El

il

X_i^l

lE_{l}^c

i_{l}c

a_i

k=l

X^k_i

k

^p

≥ 1 L^p_dEl

il

X_i^l

l

i_{l}c

a²_i 1/2

^p

≥ 1 L^p_d

sup

il

i_{l}c

a_i² 1/2

α_il:

il

Nˆ_i^l

l(α_il)≤p ^p

≥ 1 L^p_d

il

i_{l}c

a_i²

1/2

i_{l}c

x_i² 1/2

^p

≥ 1 L^p_d

i

a_ix_i ^p,

where the first inequality follows from Jensen’s inequality, the second one from hypercontractivity of chaoses gener- ated by log-concave random variables combined with the contraction principle and the third one from the induction assumption.

5. Preliminary facts

In this section we present the basic notation and tools to be used in the proof of our main results.

5.1. Some additional notation

1. Byγn,twe will denote the distribution oft Gn, whereGn=(g1, . . . , gn)is the standard Gaussian vector inRⁿ. Let alsoGⁱ_n=(gⁱ₁, . . . , g_nⁱ)be independent copies ofG_n.

2. Byνn,t we will denote the distribution oftEn, whereEn=(ξ1, . . . , ξn)is a random vector inRⁿwith independent coordinates distributed according to the symmetric exponential distribution with parameter 1. Thus ν_n,t has the density

dνn,t(x)=(2t )⁻ⁿexp

−1 t

n i=1

|x_i|

dx.

We also putEnⁱ =(ξ₁ⁱ, . . . , ξ_nⁱ)for i.i.d. copies ofEn. Let us note thatEξ_i²=2.

3. For any normαonR^n1···nd =Rⁿ¹⊗ · · · ⊗R^nd (which we will identify with the space ofd-indexed matrices, i.e.

x₁⊗ · · · ⊗x_d=(_d

j=1x^j_i

j)_{1≤i1,...,id≤n}, wherex_j=(x₁^j, . . . , x_n^j)), letρ_αbe the distance onRⁿ¹× · · · ×R^nd (which we will identify withR^n1+···+nd), defined by

ρα(x,y)=α(x1⊗ · · · ⊗xd−y1⊗ · · · ⊗yd), wherex=(x₁, . . . , x_d),y=(y₁, . . . , y_d).

Forx∈R^n1+···+nd andr≥0 letB_α(x, r)be the closed ball in the metricρ_αwith centerxand radiusr.

4. Now, forT ⊂Rⁿ¹× · · · ×R^nd,t >0, define W_d^T(α, t )=

d k=1

t^k

I⊂{1,...,d},#I=k

W_I^T(α),

where forI⊂ {1, . . . , d}, W_I^T(α)=sup

x∈T

Eα

k /∈I

x_i^k

k

k∈I

g^k_i

k

i₁,...,i_d

.

5. Similarly, fort >0,T ⊂Rⁿ¹× · · · ×R^nd we put

V_d^T(α, t ):=

d k=1

t^k

I⊂{1,...,d}: #I=k

V_I^T(α),

where

V_I^T(α):=sup

x∈T

Eα

k /∈I

x_i^k

k

k∈I

ξ_i^k

k

i₁,...,i_d

.

6. Fors, t >0,T ⊂Rⁿ¹× · · · ×R^nd, we define U_d^T(α, s, t ):=

d k=1

I,J⊂{1,...,d},

#(I∪J )=k,I∩J=∅

s^#It^#JU_I,J^T (α),

where

U_I,J^T (α):=sup

x∈T

Eα

k /∈(I∪J )

x_i^k

k

k∈I

g_i^k

k

k∈J

ξ_i^k

k

i₁,...,i_d

.

Remark. Let us notice thatU_∅^T_,I(α)=V_I^T(α),whereasU_I,^T_∅(α)=W_I^T(α).

The quantityW_I^T was defined in [15], where it played an important role in the analysis of moments of Gaussian chaoses. The quantitiesV_I^T andU_I,J^T will play an analogous role for chaoses generated by general random variables with logarithmically concave tails (as will become clear in the next section, they will allow us to bound the covering numbers for more general sets than those which were important in the Gaussian case).

5.2. Entropy estimates

In this section we present some general entropy estimates which will be crucial for bounding suprema of stochastic processes in the proof of Theorem3.2.

The first lemma we will need is a reformulation of Lemma 1 in [15]. The original statement from [15] is slightly weaker however the proof given therein justifies the version presented below.

Lemma 5.1. For any normsα₁, α₂onRⁿ,y∈B₂ⁿandt >0, γ_n,t

x: α_i(x−y)≤4tEα_i(G_n), i=1,2

≥1

2e^−1/(2t2). Lemma 5.2. For any normsα₁, α₂onRⁿ,y∈aB₁ⁿandt >0,

ν_n,t

x: α_i(x−y)≤4tEα_i(En), i=1,2

≥1 2e^−a/t. Proof. Let

K:= x∈Rⁿ: α₁(x)≤4tEα₁(En), α₂(x)≤4tEα₂(En) . By Chebyshev’s inequality,

1−νn,t(K)≤P

α₁(tEn) >4Eα₁(tEn) +P

α₂(tEn) >4Eα₂(tEn)

<1/2.

We get for anyy∈aB₁ⁿ, ν_n,t(y+K)=(2t )⁻ⁿ

K

exp

−1 t

n i=1

|x_i+y_i|

dx≥exp

−1 t

n i=1

|y_i|

K

dνn,t(x)

≥exp(−a/t )ν_n,t(K)≥1

2exp(−a/t ).

Finally, notice that ifx∈y+K, thenαi(x−y)≤4tEαi(En),i=1,2.

Before we formulate the next lemma, let us defineμn,s,t(wheres, t >0) as the convolution ofγn,s andνn,t. Lemma 5.3. For any normsα₁, α₂onRⁿ,anya >0,y∈B₂ⁿ+aB₁ⁿands, t >0,let

K= x: α₁(x−y)≤4sEα₁(G_n)+4tEα₁(En), α₂(x)≤4sEα₂(G_n)+4tEα₂(En)+α₂(y) . Then

μ_n,s,t(K)≥1

4e^{−1/(2s2)−a/t}.

Proof. We havey=y₁+y₂for somey₁∈B₂ⁿ,y₂∈aB₁ⁿ. Define K₁= x∈Rⁿ:α_i(x−y₁)≤4sEα_i(G_n), i=1,2

, K2= x∈Rⁿ:α_i(x−y2)≤4tEα_i(En), i=1,2

. Forx=x₁+x₂, wherex_j∈K_j,j=1,2,

α₁(x−y)≤α₁(x₁−y₁)+α₁(x₂−y₂)≤4sEα₁(G_n)+4tEα₁(En) and similarly

α₂(x)≤α₂(x−y)+α₂(y)≤4sEα₂(G_n)+4tEα₂(En)+α₂(y), thereforeK₁+K₂⊂K. We thus have

μn,s,t(K)≥μn,s,t(K1+K2)≥γn,s(K1)νn,t(K2)≥1

4e^{−1/(2s2)−a/t},

where in the last inequality we used Lemmas5.1and5.2.

Lemma 5.4. For anys, t >0,a=(a₁, . . . , a_d)∈(0,∞)^dandx∈(B₂ⁿ¹+a₁B₁ⁿ¹)× · · · ×(B₂^nd +a_dB₁^nd)we have μ_{n1+···+nd,s,t}

B_α

x, U_d^{x}(α,4s,4t )

≥4^−dexp

−1

2ds⁻²− a1t⁻¹

. (9)

Proof. We will proceed by induction ond. Ford =1, inequality (9) follows by Lemma5.3. Now suppose that (9) holds ford−1. We will show that it is also satisfied ford. Let us first notice that

α _d

i=1

xⁱ− d

i=1

yⁱ

≤α¹

x^d−y^d +α_yd

_d−1

i=1

xⁱ−

d−1

i=1

yⁱ

, (10)

whereα¹andα_yare norms onR^nd andR^{n1···nd−1} respectively, defined by

α¹(z):=α _d−1

i=1

xⁱ⊗z

and α_y(z):=α(z⊗y).

Then

sEα¹(Gn)+tEα¹(En)=sU_{^{_d^x_}^}_,∅(α)+t U_∅^{x_,^}_{d}(α). (11) Moreover if we putπ(x)=(x¹, . . . , x^d−1)and define a normα²_s,tonR^nd by the formula

α²_s,t(y):=U_d^{π(x)₋₁ ^}(α_y, s, t ) then

sEα_s,t² (G_n)+tEα²_s,t(En)+α_s,t² x^d

=

I,J⊂{1,...,d} I∪J=∅,I∩J=∅

s^#It^#JU_I,J^{x}(α, t )−

sU_{^{_d^x}_},∅(α)+t U_∅^{x_,^}_{d}(α)

. (12)

Notice also that by the induction assumption we have for anyz∈R^nd,

μ_{n1+···+nd−1,s,t}

y∈R^{n1+···+nd−1}: α_{z d−1}

i=1

xⁱ−

d−1

i=1

yⁱ

≤α²_4s,4t(z)

≥4^1−dexp

−(d−1)s⁻²/2−(a₁+ · · · +a_d−1)t⁻¹

. (13)

Finally let A(x):=

y∈R^n1+···+nd: α¹

x^d−y^d

≤4sEα¹(Gn_d)+4tEα¹(En_d),

α_4s,4t² y^d

≤4sEα_4s,4t² (Gn_d)+4tEα_4s,4t² (En_d)+α²_4s,4t x^d

,

α_yd

_d−1

i=1

xⁱ−

d−1

i=1

yⁱ

≤α_4s,4t² y^d

.

By (10)–(12) we getA(x)⊂Bα(x, U_d^{x}(α,4s,4t ))and therefore by (13), Lemma5.3and Fubini’s theorem we get μn₁+···+n_d,s,t

Bα

x, U_d^{x}(α,4s,4t )

≥μn₁+···+n_d,s,t

A(x)