On sharp Burkholder-Rosenthal-type inequalities for infinite-degree <span class="mathjax-formula">$U$</span>-statistics

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ON SHARP BURKHOLDER–ROSENTHAL-TYPE INEQUALITIES FOR INFINITE-DEGREE U -STATISTICS

Victor H. DE LA PEÑA^a,1, Rustam IBRAGIMOV^b,2, Shaturgun SHARAKHMETOV^c

aDepartment of Statistics, Columbia University, 2990 Broadway, New York, NY 10027, USA bDepartment of Economics, Yale University, 28 Hillhouse Ave., New Haven, CT 06511, USA cDepartment of Probability Theory, Tashkent State Economics University, Ul. Uzbekistanskaya, 49,

Tashkent, 700063, Uzbekistan

Received 15 March 2001, revised 11 March 2002

ABSTRACT. – In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degreeU-statistics. Using the approach, we prove, in particular, the following result: LetDbe the class of functionsf: R₊→R₊such that the functionf (x+z)−f (x)is concave inx∈R₊for allz∈R₊. Then the following estimate holds:

Ef _m

l=0

1i₁<···<iln

Y_i₁_,...,i_l(X_i₁, . . . , X_i_l)

m

q=0

1j1<···<jqn

Ef _m

l=q

i1<···<il−q∈{1,...,n}\{j1,...,jq}

E

Y_j₁_,...,j_q_,i₁_,...,i_l₋_q(X_j₁, . . . , X_j_q, X_i₁, . . . , X_i_l₋_q)|X_j₁, . . . , X_j_q for allf∈Dand allU-statistics

1i1<···<ilnY_i₁_,...,i_l(X_i₁, . . . , X_i_l)with nonnegative kernels Yi₁,...,i_l: R^l →R₊, 1ikn; ir =is, r=s;k, r, s=1, . . . , l; l=0, . . . , m,in independent r.v.’s X₁, . . . , Xn. Similar inequality holds for sums of decoupledU-statistics. The classD is quite wide and includes all nonnegative twice differentiable functionsf such that the function f(x)is nonincreasing inx >0,and, in particular, the power functionsf (x)=x^t, 1< t2;

the power functions multiplied by logarithm f (x)= (x+x0)^t ln(x+x0), 1< t <2, x0 max(e^(3t²⁻^6t⁺^{2)/(t (t}⁻¹⁾⁽²⁻^{t ))},1); and the entropy-type functionsf (x)=(x+x₀)ln(x +x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–

Rosenthal-type inequalities for sums ofU-statistics and prove new decoupling inequalities for E-mail addresses: vp@stat.columbia.edu (V.H. de la Peña), rustam.ibragimov@yale.edu

(R. Ibragimov), tim001@tseu.silk.org (S. Sharakhmetov).

1Supported in part by a NSF grant DMS/99/72237.

2The author thanks Victor de la Peña and the Department of Statistics, Columbia University, for their hospitality during his visits in 1999–2000.

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those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates forU-statistics of a general type.

2002 Éditions scientifiques et médicales Elsevier SAS MSC: primary 60E15; secondary 60F25

Keywords: Infinite degreeU-statistics; Burkholder–Rosenthal-type inequalities; Decoupling inequalities

RÉSUMÉ. – Dans ce travail nous présentons une méthode permettant d’obtenir certaines inégalités fines pour les espérances mathématiques de fonctions deU-statistiques de degré infini.

En particulier, nous démonstrons le résultat suivant : SoitDla classe des fonctionsf: R₊→R₊ telles quef (x+z)−f (x)est une fonction concave dex∈R₊pour chaquez∈R₊. Alors, nous avons :

Ef _m

l=0

1i₁<···<iln

Yi₁,...,i_l(Xi₁, . . . , Xi_l)

m

q=0

1j1<···<jqn

Ef _m

l=q

i₁<···<i_l₋_q∈{1,...,n}\{j₁,...,j_q}

E

Y_j₁_,...,j_q_,i₁_,...,i_l₋_q(X_j₁, . . . , X_j_q, X_i₁, . . . , X_i_l₋_q)|X_j₁, . . . , X_j_q pour toutef ∈Det touteU-statistique

1i₁<···<ilnY_i₁_,...,i_l(X_i₁, . . . , X_i_l)avec noyaux non- négatifsY_i₁_,...,i_l: R^l→R₊, 1i_k n; i_r =i_s, r=s;k, r, s=1, . . . , l;l=0, . . . , m,et variables aléatoiresX₁, . . . , X_nindépendantes. Une inégalité analogue est vraie pour les sommes deU-statistiques découplées. La classeDest assez étendue et contient toutes les fonctionsf qui sont deux fois différentiables et telles quef(x)est décroissante au sens large surx >0.

En particulier,Dcontient les fonctions puissancef (x)=x^t,1< t2 ; les fonctions puissance fois le logarithmef (x)=(x+x₀)^tln(x+x₀),1< t <2, x₀max(e^(3t²⁻^6t⁺^{2)/(t (t}⁻¹⁾⁽²⁻^{t ))},1); et les fonctions de type entropief (x)=(x+x₀)ln(x+x₀), x₀1. Comme application de ces résultats, nous détérminons les meilleures constantes dans les inégalités de type Burkholder–

Rosenthal pour les sommes desU-statistiques et nous prouvons des nouvelles inégalités de dé- couplage pour ces mêmes objets. Les résultats de ce travail sont, à notre connaissance, les pre- miers sur les meilleurs constantes dans les inégalités fines pour lesU-statistiques de type général.

2002 Éditions scientifiques et médicales Elsevier SAS

Mots Clés :U-statistiques de degré infini ; Inégalités de type Burkholder–Rosenthal ; Inégalités de découplage

1. Introduction

Recently, Klass and Nowicki [11], Ibragimov and Sharakhmetov [6–8] (see also [5]

and [9]) and Giné et al. [3] obtained Burkholder–Rosenthal-type inequalities for U- statistics with nonnegative and degenerate kernels. Ibragimov and Sharakhmetov [6]

also showed the significance of each term in the Burkholder–Rosenthal-type bounds forU-statistics of arbitrary order and obtained results concerning the rate of growth of the best constants in these bounds. Giné et al. [3] proved the Burkholder–Rosenthal- type inequalities for the tth moment of U-statistics of order m with the constants

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L^t_m(t/lnt)^mt, whereLmis a constant depending only onm, and obtained Bernstein-type exponential inequalities forU-statistics (for further discussion of the order of constants in the Burkholder–Rosenthal-type estimates for U-statistics see [6]). Ibragimov et al.

[10] found the best constants in Burkholder–Rosenthal-type inequalities for bilinear forms in the case of the fixed number of random variables (r.v.’s). de la Peña et al.

[2] determined the best constants in Burkholder–Rosenthal-type inequalities for sums of multilinear forms in independent nonnegative and symmetric r.v.’s.

In this paper, we present a method that allows one to obtain sharp inequalities for expectations of sums of U-statistics with nonnegative kernels, which represent an important case of infinite-degreeU-statistics (see [4]). Using the approach, we prove, in particular, the following Burkholder–Rosenthal-type inequality:

Ef _m

l=0

1i1<···<iln

Yi₁,...,i_l(Xi₁, . . . , Xi_l)

m

q=0

1j1<···<jqn

Ef _m

l=q

i1<···<il−q∈{1,...,n}\{j1,...,jq}

EYj₁,...,j_q,i₁,...,i_l−q(Xj₁, . . . , Xj_q, Xi₁, . . . , Xi_l−q)|Xj₁, . . . , Xj_q

for allU-statistics ₁_i₁_<_···_<i_l_nYi1,...,il(Xi1, . . . , Xil)with nonnegative kernelsYi1,...,il: R^l→R₊, 1ikn;ir =is, r=s;k, r, s=1, . . . , l;l=0, . . . , m(Yi₁,...,il ≡const0 forl=0) in independent r.v.’sX1, . . . , Xnand all functionsf: R₊→R₊such that the function f (x+z)−f (x) is concave in x∈R₊ for all z∈R₊. A similar inequality holds for sums of decoupled U-statistics. The above condition is satisfied for all twice differentiable functionsf such that the functionf(x)is nonincreasing inx >0, and, in particular, for the power functions f (x)=x^t, 1< t2; the power functions multiplied by logarithmf (x)=(x+x0)^t ln(x+x0), 1< t <2,x0max(e^(3t²⁻^6t⁺^{2)/(t (t}⁻¹⁾⁽²⁻^{t ))}, 1);

and the entropy-type functionsf (x)=(x+x0)ln(x+x0),x01.As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of regular and decoupledU-statistics with nonnegative kernels and prove new decoupling inequalities for sums ofU-statistics. We show, for instance, that the constant in the following Burkholder–Rosenthal-type inequality is sharp:

E _m

l=0

1i₁<···<i_ln

Yi₁,...,i_l(Xi₁, . . . , Xi_l) t

m

q=0

1j₁<···<jqn

E _m

l=q

i₁<···<il−q∈{1,...,n}\{j₁,...,jq}

EYj1,...,jq,i1,...,il−q(Xj1, . . . , Xjq, Xi1, . . . , Xil−q)|Xj1, . . . , Xjq

t

,

1 < t 2, for all U-statistics ₁_i₁_<_···_<i_l_nYi₁,...,il(Xi₁, . . . , Xil) with nonnegative kernelsYi₁,...,il: R^l→R₊, 1ikn;ir =is, r=s;k, r, s=1, . . . , l;l=0, . . . , m, in

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independent r.v.’sX1, . . . , Xn.A similar result holds for sums of decoupledU-statistics.

To our knowledge, the results obtained in the paper are the first known results on the best constants in sharp two-sided moment estimates forU-statistics of a general type.

2. Sharp estimates for expectations of functions of sums ofU-statistics Let R₊ = [0,∞), 1 m n, X1, . . . , Xn, Xp1, . . . , Xpn, p = 1, . . . , m, be independent r.v.’s and let Yi₁,...,i_l: R^l → R₊, 1 ik n; ir = is, r = s; k, r, s = 1, . . . , l; l = 0, . . . , m, be functions having the property that Yi1,...,il(x1, . . . , xl) = Yi_π(1),...,i_π(l)(xπ(1), . . . , xπ(l)), xk ∈R, k=1, . . . , l, 1i1<· · ·< il n, for all permu- tations π:{1, . . . , l} → {1, . . . , l},l=2, . . . , m(we assume that Yi1,...,il ≡const0 for l=0). Consider the sums of regularU-statistics (symmetric statistics)

m

l=0

1i₁<···<i_ln

Yi1,...,il(Xi1, . . . , Xil)

and decoupled U-statistics (symmetric statistics) m

l=0

1i1<···<iln

Yi₁,...,il(X1,i₁, . . . , Xl,il).

In what follows, write

Y^reg(i1, . . . , il)=Yi1,...,il(Xi1, . . . , Xil), Y^dec(i1, . . . , il)=Yi1,...,il(X1,i1, . . . , Xl,il).

Denote by D the class of functions f: R₊ → R₊ such that the function f (x + z)−f (x) is concave in x ∈ R₊ for all z ∈ R₊. The class D is quite wide and includes all nonnegative twice differentiable functionsf such that the functionf(x)is nonincreasing inx >0, and, in particular, the power functionsf1(x)=x^t, 1< t2; the power functions multiplied by logarithm f2(x)=(x+x0)^t ln(x+x0), 1< t <2,x0 max(e^(3t²⁻^6t⁺^{2)/(t (t}⁻¹⁾⁽²⁻^{t ))}, 1), and the entropy-type functions f3(x)=(x+x0)ln(x+ x0),x01.Indeed, if the functionf(x)is nonincreasing inx >0, then we havef(x+ z)f(x)for allx >0,z0, and, therefore,f (x+z)−f (x)is concave inx∈R₊for allz∈R₊.It is obvious thatf₁(x)is nonincreasing inx >0 and, therefore,f1∈D.In addition to that,f₂(x)=(x+x0)^t⁻³(t (t−1)(t−2)ln(x+x0)+3t²−6t+2)0,x >0, and, therefore,f₂(x)is nonincreasing inx >0, andf2∈D.Sincef₃(x)=1/(x+x0) is nonincreasing inx >0, we havef3∈D.

In the inequalities throughout the paper, the extremal cases of the estimates such as +∞ +∞ are considered to be valid inequalities; we, therefore, do not include assumptions on finiteness of moments of the summand r.v.’s that ensure finiteness of moments of sums ofU-statistics into formulations of the results.

The following theorems give sharp Burkholder–Rosenthal-type inequalities for sums ofU-statistics. In what follows,E(· |Xj1, . . . , Xjq)=E(· |Xj1,ij1, . . . , Xjq,i_jq)=E(·), the unconditional expectation operator, forq=0.

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THEOREM 1. – Forf ∈D, Ef

_m

l=0

1i₁<···<i_ln

Y^reg(i1, . . . , il)

m

q=0

1j₁<···<j_qn

Ef _m

l=q

i₁<···<i_l−q∈{1,...,n}\{j₁,...,jq}

EY^reg(j1, . . . , jq, i1, . . . , il−q)|Xj₁, . . . , Xjq

. (1)

THEOREM 2. – Forf ∈D, Ef

_m

l=0

1i1<···<iln

Y^dec(i1, . . . , il)

m

q=0

1j1<···<jqm

1ij1<···<i_jqn

Ef _m

l=j_q

1ipn,ip1<ip2, p1<p2, p,p1,p2=1,...,l, p=j1,...,jq

EY^dec(i1, . . . , il)|Xj₁,i_j

1, . . . , Xj_q,i_jq

. (2)

COROLLARY 1. – For a twice differentiable function f: R₊ →R₊ such that the functionf(x)is nonincreasing onx >0, inequalities (1) and (2) hold.

THEOREM 3. – The constants in the following inequalities are sharp:

E _m

l=0

1i₁<···<i_ln

Y^reg(i1, . . . , il) t

m

q=0

1j₁<···<j_qn

E _m

l=q

i₁<···<i_l−q∈{1,...,n}\{j₁,...,j_q}

EY^reg(j1, . . . , jq, i1, . . . , il−q)|Xj₁, . . . , Xjq

t

, (3)

E _m

l=0

1i₁<···<iln

Y^dec(i1, . . . , il) t

m

q=0

1j₁<···<jqm

1ij1<···<i_jqn

E _m

l=jq

1ipn, ip1<ip2, p₁<p₂, p,p1,p2=1,...,l, p=j1,...,jq

EY^dec(i1, . . . , il)|Xj₁,i_j

1, . . . , Xj_q,i_jq

t

, 1< t2. (4)

Remark 1. – It is not difficult to see that moment inequalities (2) and (4) for sums of decoupled U-statistics follow from their counter-parts (1) and (3) for sums of

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regular U-statistics, using the fact that any decoupled U-statistic can be represented as an undecoupled U-statistic with many zero kernels (it suffices to consider new r.v.’sX˜(p−1)n+i =Xpi,p=1, . . . , m,i=1, . . . , n, and new kernelsY˜i1,n+i2,...,(l−1)n+il = Yi₁,i₂,...,il, 1i1< i2<· · ·< il n, l=0, . . . , m;Y˜j₁,j₂,...,jl =0, 1j1< j2<· · ·<

jl mn,(j1, j2, . . . , jl)=(i1, n+i2, . . . , (l−1)n+il)for 1i1< i2<· · ·< iln, l=1, . . . , m;Yjπ(1),...,jπ(l)(x1, . . . , xl)=Yj1,...,jl(x_π−1(1), . . . , x_π−1(l)),xk ∈R,k=1, . . . , l, 1j1<· · ·< jl mn, for all permutations π:{1, . . . , l} → {1, . . . , l}, l=2, . . . , m, whereπ⁻¹is the inverse ofπ).

Remark 2. – The essence of the Burkholder–Rosenthal-type bounds for sums ofU- statistics given by Theorems 1–3 is that they give (sharp) estimates for moments of the sums in terms of expressions that do not contain moments of sums of r.v.’s. The bounds contain only directly computable expressions. For example, in the case of regular U- statistics of ordermin identically distributed r.v.’s the bounds consist of terms equivalent to n^(m⁻^k)t⁺^kE(E(Y^reg(X1, . . . , Xm)|X1, . . . , Xk))^t, k=0,1, . . . , m (and each of the terms is significant, as it was shown in [6], see also [5]). In the case of, let us say, sums of multilinear forms the terms in the bounds depend only on the moments of individual variables (see also [2]).

Remark 3. – From the results obtained in [5–9,11] (see also [3]) it follows that the following non-sharp (in the sense of constants) Burkholder–Rosenthal-type inequality holds for regular U-statistics of second order with nonnegative kernels (below,Ci(t), C_i^reg(t)andC_i^dec(t)are constants depending ontonly):

E

1i<jn

Y_ij^reg(Xi, Xj) t

C1(t)

1i<jn

EY_ij^reg(Xi, Xj)^t

+C2(t)

n−1

i=1

E _n

j=i+1

EY_ij^reg(Xi, Xj)|Xi

t

+C3(t) n

j=2

E _j₋₁

i=1

EY_ij^reg(Xi, Xj)|Xj

t

+C4(t)

1i<jn

EY_ij^reg(Xi, Xj) t

, t >1.

From (3) it follows that a “natural” form of Burkholder–Rosenthal-type inequality for regularU-statistics of second order with nonnegative kernels contains three, but not four terms and is given by

E

1i<jn

Y_ij^reg(Xi, Xj) t

C₁^reg(t)

1i<jn

EY_ij^reg(Xi, Xj)^t +C₂^reg(t)

n

i=1

E

j=i

EY_ij^reg(Xi, Xj)|Xi

^t

+C₃^reg(t)

1i<jn

EY_ij^reg(Xi, Xj) t

.

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Moreover, the best constants in the inequality are given byC_i^reg(t)=1,i=1,2,3, for 1< t2.Similarly, from (4) it follows that a “natural” form of Burkholder–Rosenthal- type inequality for decoupled U-statistics of second order with nonnegative kernels contains four terms similar to those in [8], namely,

E

1i<jn

Y_ij^dec(X1i, X2j) t

C₁^dec(t)

1i<jn

EY_ij^dec(X1i, X2j)^t

+C₂^dec(t)

n−1

i=1

E _n

j=i+1

EY_ij^dec(X1i, X2j)|X1i

t

+C₃^dec(t) n

j=2

E _j₋₁

i=1

EY_ij^dec(X1i, X2j)|X2j

t

+C₄^dec(t)

1i<jn

EY_ij^reg(X1i, X2j) t

,

and, moreover, the best constants in the above inequality are given by C_i^dec(t)=1, i=1,2,3,4, for 1< t2.

Remark 4. – Similarly to Remark 3, from moment inequalities for sums of multilinear forms obtained by Peña et al. [2] and Theorems 1–3 it follows that a “natural” form of Burkholder–Rosenthal-type inequalities for expectations of functions of sums of regular U-statistics of order not greater thanmwith nonnegative kernels containsm+1 terms and a “natural” form of Burkholder–Rosenthal-type inequalities for expectations of functions of sums of decoupledU-statistics of order not greater thanmwith nonnegative kernels contains 2^mterms. Moreover, those theorems imply the following inequalities:

E _m

l=0

1i₁<···<iln

Y^reg(i1, . . . , il) t

(m+1) max

q=0,...,m

1j1<···<jqn

E _m

l=q

i₁<···<i_l−q∈{1,...,n}\{j₁,...,j_q}

EY^reg(j1, . . . , jq, i1, . . . , il−q)|Xj₁, . . . , Xj_q

t

,

E _m

l=0

1i₁<···<i_ln

Y^dec(i1, . . . , il) t

2^m max

q=0,...,m max

1j1<···<jqm

1i_j

1<···<i_jqn

E _m

l=jq

1i_pn, i_p

1<i_p

2, p₁<p₂, p,p₁,p₂=1,...,l, p=j₁,...,j_q

EY^dec(i1, . . . , il)|Xj1,ij1, . . . , Xjq,i_jq

t

,

1< t2.

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From the estimate N

k=1

z^t_k _N

k=1

zk

t

, z1, . . . , zN0, t >1, (5)

and Jensen’s inequality it follows that E

_m

l=0

1i₁<···<iln

Y^reg(i1, . . . , il) t

max

q=0,...,m

1j1<···<jqn

E _m

l=q

i1<···<il−q∈{1,...,n}\{j1,...,jq}

EY^reg(j1, . . . , jq, i1, . . . , il−q)|Xj₁, . . . , Xj_q

t

, (6)

E _m

l=0

1i₁<···<i_ln

Y^dec(i1, . . . , il) t

max

q=0,...,m max

1j1<···<jqm

1i_j

1<···<i_jqn

E _m

l=jq

1i_pn, i_p₁<i_p₂, p₁<p₂, p,p₁,p₂=1,...,l, p=j₁,...,j_q

EY^dec(i1, . . . , il)|Xj₁,ij1, . . . , Xjq,i_jq

t

, (7)

1< t 2. Assume that X_p1 , . . . , X_pn, p=1, . . . , m, are independent copies of the r.v.’sX1, . . . , Xn (the primes are used to remind us about the independence between the sequences). From estimate (5), the inequality (^N_k₌₁zk)^t N^t⁻¹^N_k₌₁z^t_k,z1, . . . , zN 0, t >1, and estimates (3), (4), (6) and (7) it follows that the following theorem holds (C_m^k =m!/(k!(m−k)!), 0km).

THEOREM 4. – The following decoupling inequalities hold:

(m+1)⁻¹E _m

l=0

1i1<···<iln

Yi₁,...,il(X_1,i ₁, . . . , X_l,i _l) t

E

_m

l=0

1i1<···<iln

Yi₁,...,il(Xi₁, . . . , Xil) t

_m

k=0

C_m^k^t

E _m

l=0

1i₁<···<i_ln

Yi1,...,il(X_1,i₁, . . . , X_l,i_l) t

, 1< t2.

Note that the constant in the upper decoupling inequality given by Theorem 4 satisfies the inequality ^m_k₌₀(C_m^k)^t 2^mt.As far as we know, the constants in the estimates in Theorem 4 are the best available so far, and it is likely that they are the sharp ones.