Asymptotics for transportation cost in high dimensions

LetX ₁,...,X _n ,Y ₁,...,Y _n be i.i.d. with the law μ on the cube [0, 1] ^d ,d?3. LetL _n (μ)=inf_πΣ _i=1 ⁿ ||X _i ?Y _π(i)|| denote the optimal bipartite matching of theX andY points, where π ranges over all permutations of the integers 1, 2,...,n, and where ‖·‖ is a norm on ? ^d . If μ is Lebesgue measure it is shown that $$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha {\text{a}}{\text{.s}}{\text{.}}$$ where α is a finite constant depending on ‖ ‖ andd only. More generally, for arbitrary μ it is shown that $$\mathop {\lim }\limits_{n \to \infty } L_n (\mu )/n^{(d - 1)/d} = \alpha \int {(f{\text{(}}x{\text{)}})^{(d - 1)/d} dxa.s.} $$ wheref is the density of the absolutely continuous part of μ. We also find the rate of convergence.     

The Continuity of M and N in Greedy Lattice Animals

Let {X _v: v ∈ Z ^d}, d≥2, be i.i.d. positive random variables with the common distribution F which satisfy, for some a>0, ∫ x ^d (log⁺ x) ^d+a dF(x)<∞ Define $$M_n = \max \left\{ {\sum\limits_{\upsilon \in \pi } {X_\upsilon } {\kern 1pt} :\pi {\text{ a selfavoiding path of length }}n{\text{ starting at the origin}}} \right\}$$ $$N_n = \max \left\{ {\sum\limits_{\upsilon \in \xi } {X_\upsilon } {\kern 1pt} :\xi {\text{ a lattice animal of size }}n{\text{ containing the origin}}} \right\}$$ Then it has been shown that there exist positive finite constants M = M[F] and N = N[F] such that $${\mathop {\lim }\limits_{n \to \infty }} \frac{{M_n }}{n} = M{\text{ and }}{\mathop {\lim }\limits_{n \to \infty }} \frac{{N_n }}{n} = N{\text{ a}}{\text{.s}}{\text{. and in }}L^1 $$      

On the strong summation of Fourier series with variable exponents

Пустьf 2π-периодическ ая суммируемая функц ия, as _k (x) еë сумма Фурье порядк аk. В связи с известным ре зультатом Зигмунда о сильной суммируемости мы уст анавливаем, что если λn→∞, то сущес твует такая функцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _{2n} } } \right\}^{1/\lambda _{2n} } = \infty .$$ Отсюда, в частности, вы текает, что если λn?∞, т о существует такая фун кцияf, что почти всюду $$\mathop {\lim \sup }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } } \right\}^{1/\lambda _n } = \infty .$$ Пусть, далее, ω-модуль н епрерывности и $$H^\omega = \{ f:\parallel f(x + h) - f(x)\parallel _c \leqq K_f \omega (h)\} .$$ . Мы доказываем, что есл и λ _n ?∞, то необходимым и достаточным условие м для того, чтобы для всехf∈H ^ω выполнялос ь соотношение $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\frac{1}{n}\mathop \sum \limits_{k = n + 1}^{2n} |s_k (x) - f(x)|^{\lambda _n } } \right\}^{1/\lambda _n } = 0(x \in [0;2\pi ])$$ является условие $$\omega \left( {\frac{1}{n}} \right) = o\left( {\frac{1}{{\log n}} + \frac{1}{{\lambda _n }}} \right).$$ Это же условие необхо димо и достаточно для того, чтобы выполнялось соотнош ение $$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{n + 1}}\mathop \sum \limits_{k = 0}^n |s_k (x) - f(x)|^{\lambda _k } = 0(f \in H^\omega ,x \in [0;2\pi ]).$$      

Lipschitz-Nikolski\imath constants and asymptotic simultaneous approximation on theM n -operatorsconstants and asymptotic simultaneous approximation on theM n -operators

This note is a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by theM _n -operators of Meyer-König and Zeller which are defined by $$(M_n f)(x) = (1 - x)^{n + 1} \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{{n + k}}} \right)} \left( \begin{array}{l} n + k \\ k \\ \end{array} \right)x^k , n = 1,2,....$$ Among other results it is proved that for 0<α≤1 $$\mathop {\lim }\limits_{n \to \infty } n^{\alpha /2} \mathop {\sup }\limits_{f \in Lip_1 \alpha } \left| {(M_n f)(x) - f(x)} \right| = \frac{{\Gamma \left( {\frac{{\alpha + 1}}{2}} \right)}}{{\pi ^{1/2} }}\left\{ {2x(1 - x)^2 } \right\}^{\alpha /2} $$ and if for a functionf, the derivativeD ^m+2 f exist at a pointx∈(0, 1), then $$\mathop {\lim }\limits_{n \to \infty } 2n[D^m (M_n f) - D^m f] = \Omega f,$$ where Ω is the linear differential operator given by $$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$      

On the convergence to infinity of Fourier series along dense subsequences of numbers

В статье доказываетс я Теорема.Какова бы ни была возрастающая последовательность натуральных чисел {H _k } _{k = 1} ^∞ c $$\mathop {\lim }\limits_{k \to \infty } \frac{{H_k }}{k} = + \infty$$ , существует функцияf∈L(0, 2π) такая, что для почт и всех x∈(0, 2π) можно найти возраст ающую последовательность номеров {n_k(x)} _k=1 ^∞ ,удовлетворяющую усл овиям 1) $$n_k (x) \leqq H_k , k = 1,2, ...,$$ 2) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t} (x)} (x,f) = + \infty ,$$ 3) $$\mathop {\lim }\limits_{t \to \infty } S_{n_{2t - 1} (x)} (x,f) = - \infty$$ .     

Distribution of points for convergent interpolatory integration rules on (−∞, ∞)

Letw be a “nice” positive weight function on (?∞, ∞), such asw(x)=exp(??x?^α) α>1. Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$ is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1, $$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$ Moreover, suppose that the sequence of rules {I _n} _n=1 ^t8 isconvergent: $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - \infty }^\infty {f(x)w(x)dx} $$ for all continuousf:R→R satisfying suitable integrability conditions. What then can we say about thedistribution of the points {x _jn} _j=1 ⁿ ,n≥1? Roughly speaking, the conclusion of this paper is thathalf the points are distributed like zeros of orthogonal polynomials forw, and half may bearbitrarily distributed. Thus half the points haveNevai-Ullmann distribution of order α, and the rest are arbitrarily distributed. We also describe the possible distributions of the integration points, when the ruleI _n has precision other thann-1.     

Pointwise and uniform estimates for convex approximation of functions by algebraic polynomials

Let Δ ^q be the set of functionsf for which theqth difference, is nonnegative on the interval [? 1,1],P _n is the set of algebraic polynomials of degree not exceedingn, τ _k (f, δ) _p is the averaged Sendov-Popov modulus of smoothness in theL _p [?1,1] metric for 1≦p≦∞, ω _k (f, δ) and $\omega _\phi ^k (f,\delta ),\phi (x): = \sqrt {1 - x^2 } ,$ , are the usual modulus and the Ditzian-Totik modulus of smoothness in the uniform metric, respectively. For a functionf∈C[?1,1]?Δ² we construct a polynomialp _n ∈P _n ?Δ² such that $$\begin{gathered} \left| {f(x) - p_n (x)} \right| \leqslant C\omega _3 (f,n^{ - 1} \sqrt {1 - x^2 } + n^{ - 2} ),x \in [ - 1,1]; \hfill \\ \left\| {f - p_n } \right\|_\infty \leqslant C\omega _\phi ^3 (f,n^{ - 1} ); \hfill \\ \left\| {f - p_n } \right\|_p \leqslant C\tau _3 (f,n^{ - 1} )_p . \hfill \\ \end{gathered}$$ As a consequence, for a functionf∈C ²[?1,1]?Δ³ a polynomialp _n ^* ∈P _n ?Δ³ exists such that $$\left\| {f - p_n^* } \right\|_\infty \leqslant Cn^{ - 1} \omega _2 (f\prime ,n^{ - 1} ),$$ wheren≥2 andC is an absolute constant.     

Rational approximations of convex functions

The following inequalities are shown to hold for the least uniform rational deviations R_n(f) of a function f(x), continuous and convex in the interval [a, b]: $$R_n (f) \leqslant C(v)\Omega (f)n^{ - 1} \overbrace {\ln \ldots \ln }^{vtimes}n$$ (ν is an integer, C(ν) depends only on ν, and Ω(f) is the total oscillation of f); $$R_n (f) \leqslant C_1 n^{ - 1} \overbrace {\ln \ldots \ln }^{vtimes}n\mathop {\inf }\limits_{(b - a)\chi _n \leqslant \lambda< b - a} \left\{ {\omega (\lambda ,f) + M(f)n^{ - 1} \ln \frac{{b - a}}{\lambda }} \right\}$$ (ν is an integer, C₁(ν) depends only on ν, x_n = exp (-n/(500 In²n)), ω (δ,f) is the modulus of continuity of f, and M(f) = max¦f(x) ¦.     

On the divergence of de la Vallée Poussin means of Fourier series

Пусть {λ _{n 1} ^t8 — монотонн ая последовательнос ть натуральных чисел. Дл я каждой функции fεL(0, 2π) с рядом Фурье строятся обобщенные средние Bалле Пуссена $$V_n^{(\lambda )} (f;x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{k = 1}^n (a_k \cos kx + b_k \sin kx) + \mathop \sum \limits_{k = n + 1}^{n + \lambda _n } \left( {1 - \frac{{k - n}}{{\lambda _n + 1}}} \right)\left( {a_k \cos kx + b_k \sin kx} \right).$$ Доказываются следую щие теоремы.

1. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность {V_n ^(λ)(?;x)} расходится почти вс юду.
2. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность $$\left\{ {\frac{1}{\pi }\mathop \smallint \limits_{ - \pi /\lambda _n }^{\pi /\lambda _n } f(x + t)\frac{{\sin (n + \tfrac{1}{2})t}}{{2\sin \tfrac{1}{2}t}}dt} \right\}$$ расходится почти всю ду

.     

A reciprocity formula for quadratic forms

LetQ(x) denote a quadratic form over the rational integers in four variables (x=(x₁,...,x₄)). ThenQ is representable as a symmetric matrix. Assume this matrix to be non-singular modp(p≠2 prime); then the “inverse” quadratic formQ ^?1 modp can be defined. Letf:?⁴→? be defined such that the Fourier transformf exists and the sum $$\sum\limits_{x \in \mathbb{Z}^4 } {f(c x), c \in \mathbb{R}, c \ne 0} $$ is convergent. Furthermore, letm=p ₁...p _k be the product ofk distinct primes withm>1, 2×m; let $$\varepsilon = \prod\limits_{i = 1}^k {\left( {\frac{{\det Q}}{{p_i }}} \right)} \ne 0$$ for the Legendre symbol $$\left( {\frac{ \cdot }{p}} \right)$$ ; define $$B_i (Q,x) = \left\{ {\begin{array}{*{20}c} {1 for Q(x) \equiv 0\bmod p_i } \\ , \\ {0 for Q(x)\not \equiv 0\bmod p_i } \\ \end{array} } \right.$$ and forr∈?,r>0, $$F(Q,f,r) = \sum\limits_{x \in \mathbb{Z}^4 } {\left( {\prod\limits_{i = 1}^k {\left( {B_i (Q,x) - \frac{1}{{p_i }}} \right)} } \right)f(r^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} x)} $$ Then we have $$F(Q,f,m) = \varepsilon F(Q^{ - 1} ,\hat f,m)$$      

On the approximation by de la Vallée Poussin sums and interpolatory polynomials in Lipschitz norms

ПустьM ^m,α - множество 2π-п ериодических функци йf с конечной нормой $$||f||_{p,m,\alpha } = \sum\limits_{k = 1}^m {||f^{(k)} ||_{_p } + \mathop {\sup }\limits_{h \ne 0} |h|^{ - \alpha } ||} f^{(m)} (o + h) - f^{(m)} (o)||_{p,} $$ где1 ≦ p ≦ ∞, 0≦α≦1. Рассмотр им средние Bалле Пуссе на $$(\sigma _{n,1} f)(x) = \frac{1}{\pi }\int\limits_0^{2x} {f(u)K_{n,1} (x - u)du} $$ и $$(L_{n,1} f)(x) = \frac{2}{{2n + 1}}\sum\limits_{k = 1}^{2n} {f(x_k )K_{n,1} } (x - x_k ),$$ де0≦l≦n и x _k=2kπ/(2n+1). В работе по лучены оценки для вел ичин \(||f - \sigma _{n,1} f||_{p,r,\beta } \) и $$||f - L_{n,1} f||_{p,r,\beta } (r + \beta \leqq m + \alpha ).$$      

On some properties of trigonometric series with monotone decreasing coefficients

В статье рассматрива ются одномерные и дву мерные тригонометрические ряды с моно-тонными коэффициентами. Дает ся пример двойного тригонометрическог о ряда (1) $$\mathop \sum \limits_{n,k = 1}^\infty a_{nk} \sin nx\sin ky,$$ , коэффициенты которо го монотонны поk и поп, любая последовательность \(\{ S_{n_k m_k } (x,y)\} _{k = 1}^\infty\) прямоугольных части чных сумм ряда (1), где min(n _k ,m _k )→∞ приk→∞, расходится по чти всюду на (0,n)². Кроме того, изучается мера множеств нулей ф ункций (2) $$f(x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{n = 1}^{a_0 } a_n \cos nx\tilde f(x) = \mathop \sum \limits_{n = 1}^\infty a_n \sin nx,$$ , гдеа _n ↓ приn→ ∞, и доказ ьшается несколько те орем о скорости убывания ко эффициентовa _n рядов (2), если все част ичные суммыS _n (f,x) или \(S_n (\tilde f,x)\) дляn=1,2,... неотрицате ль-ны на (0,n).     

Curvature estimates for submanifolds with prescribed Gauss image and mean curvature

We study that the n-graph defined by a smooth map ${f:\Omega\subset\mathbb R^{n}\to \mathbb R^{m}, m\ge 2,}$ in ${\mathbb R^{m+n}}$ of the prescribed mean curvature and the Gauss image. Under the condition $$\Delta_f=\left[\text{det}\left(\delta_{ij}+\sum_\alpha{\frac {\partial {f^\alpha}}{\partial {x^i}}}{\frac {\partial {f^\alpha}}{\partial {x^j}}}\right)\right]^{\frac{1}{2}} < 2,$$ we derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le{\frac{C}{R^2}}$$ when 2 ≤ n ≤ 5 with constant C depending on the given geometric data. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le CR^{-a}\sup_{D_{2R}(x)}(2-\Delta_f)^{-\left({\frac{3}{2}}+{\frac{1}{s}}\right)},\quad s=\min(m, n)$$ with a < 1. If the image under the Gauss map is contained in a geodesic ball of the radius ${{\frac{\sqrt{2}}{4}}\pi}$ in G _n,m we also derive corresponding estimates.     

On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).     

An analog of the weyl decomposition of the spaceL q (ω, ℝ n ) for a first-order differential operatorfor a first-order differential operator

We prove the following theorem: Suppose the function f(x) belongs toL _q (ω, ? ⁿ ), ω ? ? ^m , q∈(1, ∞), and satisfies the inequality $$|\int\limits_\omega {(f(x),{\mathbf{ }}v(x)){\mathbf{ }}dx| \leqslant \mu ||} v||'_q ,{\mathbf{ }}\tfrac{1}{q} + \tfrac{1}{{q'}} = 1,$$ for all n-dimensional vector-valued functions in the kernel of a scalar-valued first-order differential operator £ for which the second-order operatorLL ^* is elliptic. Then there exists a function p(x)∈W _q ¹ (ω) such that $$||f(x) - \mathfrak{L}^* p(x)||q \leqslant C_q \mu .$$ Bibliography: 6 titles.     

Approximation of integrable functions by linear methods almost everywhere

It is shown that 2π periodic functions whose (r-1)-th derivatives have bounded variation (r > 0) can be approximated by de La Vallée-Poussin sums σ _n,m (an ?m =m (n) ?An,0 <a<A<1) at almost all points with a rate o(n^?r). For functions belonging to the class Lip (α, L) (0 <α < 1), any natural N, and a positive ?, we have almost everywhere $$|f(x) - \sigma _{n,m} (f;x)| \leqslant c(f,x)n^{ - \alpha } lnn \ldots ln_N^{1 + \varepsilon } n,$$ where \(ln_k x = \underbrace {ln \ldots ln x}_k(k = 1, 2, \ldots )\) . For any triangular method of summation T with bounded coefficients we construct functions belonging to Lip (α, L) (0 < α < 1) and such that almost everywhere, $$\mathop {\overline {\lim } }\limits_{n \to \infty } |f(x) - \tau _n (f;x)|n^a (ln n \ldots ln_N n)^{ - a} = \infty $$ where the τ_n(f; x) are the means of the method T.     

Inequalities for the Volume of the Unit Ball in {\mathbb{R}^{n}}

The volume of the unit ball in ${\mathbb{R}^{n}}$ is defined by $$\Omega_{n} = \frac{\pi^{n/2}}{\Gamma(n/2+1)},\qquad n = 1,2,3,\ldots,$$ where Γ denotes the classical gamma function of Euler. In several recently published papers numerous authors studied properties of Ω _n . In particular, various inequalities involving Ω _n are given in the literature. In this paper, we continue the work on this subject and offer new inequalities. More precisely, we offer sharp upper and lower bounds for $$\frac{\Omega_{n}^{2}}{\Omega_{n-1} \Omega_{n+1}},\quad\frac{\Omega_{n}}{\Omega_{n-1}+\Omega_{n+1}} \quad {\rm and} \quad\Omega_{n}.$$      

Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

Let {X _k,i ; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {p _n ; n ≥ 1} be a sequence of positive integers such that n/p _n is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry ${L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}$ of the sample correlation matrix ${{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}$ where ${\hat{\rho}^{(n)}_{i,j}}$ denotes the Pearson correlation coefficient between (X _1,i , ..., X _n,i )′ and (X _1,j ,...,X _n,j )′. Write ${F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}$ , ${W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}$ , and ${W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}$ . Under the assumption that ${\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}$ for some δ > 0, we show that the following six statements are equivalent: $$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$ $$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$ $$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$ $$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$ $$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$ $$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$ where ${\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}$ , and a _n  = 4 log p _n ? log log p _n . The equivalences between (i), (ii), (iii), and (v) assume that only ${\mathbb{E}X_{1,1}^{2} < \infty}$ . Weak laws of large numbers for W _n and L _n , n ≥  1, are also established and these are of the form ${W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)$ and ${n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)$ , respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L _n obtained by Jiang. Some open problems are also posed.     

On the uniform complete convergence of estimates for multivariate density functions and regression curves

Let (X ₁,Y ₁),...(X _n ,Y _n ) be a random sample from the (k+1)-dimensional multivariate density functionf ^*(x,y). Estimates of thek-dimensional density functionf(x)=∫f ^*(x,y)dy of the form $$\hat f_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n W \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$ are considered whereW(x) is a bounded, nonnegative weight function andb ₁ (n),...,b _k (n) and bandwidth sequences depending on the sample size and tending to 0 asn→∞. For the regression function $$m(x) = E(Y|X = x) = \frac{{h(x)}}{{f(x)}}$$ whereh(x)=∫y(f) ^* (x, y)dy , estimates of the form $$\hat h_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n {Y_i W} \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$ are considered. In particular, unform consistency of the estimates is obtained by showing that \(||\hat f_n (x) - f(x)||_\infty \) and \(||\hat m_n (x) - m(x)||_\infty \) converge completely to zero for a large class of “good” weight functions and under mild conditions on the bandwidth sequencesb _k (n)'s.