共查询到20条相似文献,搜索用时 15 毫秒
1.
K. F. Cheng 《Annals of the Institute of Statistical Mathematics》1982,34(1):479-489
Summary Letf
n
(p)
be a recursive kernel estimate off
(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of
and show that the rate of almost sure convergence of
to zero isO(n
−α), α<(r−p)/(2r+1), iff
(r),r>p≧0, is a continuousL
2(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of
to zero under different conditions onf.
This work was supported in part by the Research Foundation of SUNY. 相似文献
2.
张涤新 《中国科学A辑(英文版)》2002,45(2):223-232
Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.`` 相似文献
3.
Z. Xu 《Acta Mathematica Hungarica》2010,127(4):301-319
Let f be a primitive positive integral binary quadratic form of discriminant −D, and r
f
(n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of $
\sum\limits_{n \leqq x} {r_f (n)^\beta }
$
\sum\limits_{n \leqq x} {r_f (n)^\beta }
for any positive integer β. Next, we give an estimate of ∫1
T
|E(x)|2
x
−3/2
dx when β = 1. 相似文献
4.
We consider the problem of nonparametric identification for a multi-dimensional functional autoregression y
t
= f(y
t
−1, …,y
t−d
) + e
t
on the basis of N observations of y
t
. In the case when the unknown nonlinear function f belongs to the Barron class, we propose an estimation algorithm which provides approximations of f with expected L
2 accuracy O(N
1/4ln1/4
N). We also show that this approximation rate cannot be significantly improved.
The proposed algorithms are “computationally efficient”– the total number of elementary computations necessary to complete
the estimate grows polynomially with N.
Received: 23 September 1997 / Revised version: 28 January 1999 相似文献
5.
Assume thatf is an integer transcendental solution of the differential equationP
n
(z, f, f′)=P
n−1(z, f, f′, ... f
(p)), whereP
n
andP
n−1 are polynomials in all variables, the degree ofP
n
with respect tof andf′ is equal ton, and the degree ofP
n−1 with respect tof, f′, ... f
(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E
*, whereE
* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z
1/2 (β+o(1)), β∈C, holds forz=re
iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re
iν)‖=o(r
1/2),r→+∞,r>0,
, where Δ is a certain set on the semiaxisr>0 with mes Δ<∞.
“L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77,
January, 1999. 相似文献
6.
An (n, d, k)-mapping f is a mapping from binary vectors of length n to permutations of length n + k such that for all x, y
{0,1}n, dH (f(x), f(y)) ≥ dH (x, y) + d, if dH (x, y) ≤ (n + k) − d and dH (f(x), f(y)) = n + k, if dH (x, y) > (n + k) − d. In this paper, we construct an (n,3,2)-mapping for any positive integer n ≥ 6. An (n, r)-permutation array is a permutation array of length n and any two permutations of which have Hamming distance at least r. Let P(n, r) denote the maximum size of an (n, r)-permutation array and A(n, r) denote the same setting for binary codes. Applying (n,3,2)-mappings to the design of permutation array, we can construct an efficient permutation array (easy to encode and decode)
with better code rate than previous results [Chang (2005). IEEE Trans inf theory 51:359–365, Chang et al. (2003). IEEE Trans
Inf Theory 49:1054–1059; Huang et al. (submitted)]. More precisely, we obtain that, for n ≥ 8, P(n, r) ≥ A(n − 2, r − 3) > A(n − 1,r − 2) = A(n, r − 1) when n is even and P(n, r) ≥ A(n − 2, r − 3) = A(n − 1, r − 2) > A(n, r − 1) when n is odd. This improves the best bound A(n − 1,r − 2) so far [Huang et al. (submitted)] for n ≥ 8.
The work was supported in part by the National Science Council of Taiwan under contract NSC-93-2213-E-009-117 相似文献
7.
Asymptotic Upper Bounds for Ramsey Functions 总被引:5,自引:0,他引:5
We show that for any graph G with N vertices and average degree d, if the average degree of any neighborhood induced subgraph is at most a, then the independence number of G is at least Nf
a
+1(d), where f
a
+1(d)=∫0
1(((1−t)1/(
a
+1))/(a+1+(d−a−1)t))dt. Based on this result, we prove that for any fixed k and l, there holds r(K
k
+
l
,K
n
)≤ (l+o(1))n
k
/(logn)
k
−1. In particular, r(K
k
, K
n
)≤(1+o(1))n
k
−1/(log n)
k
−2.
Received: May 11, 1998 Final version received: March 24, 1999 相似文献
8.
H. A. Dzyubenko 《Ukrainian Mathematical Journal》2009,61(4):519-540
In the case where a 2π-periodic function f is twice continuously differentiable on the real axis ℝ and changes its monotonicity at different fixed points y
i
∈ [− π, π), i = 1,…, 2s, s ∈ ℕ (i.e., on ℝ, there exists a set Y := {y
i
}
i∈ℤ of points y
i
= y
i+2s
+ 2π such that the function f does not decrease on [y
i
, y
i−1] if i is odd and does not increase if i is even), for any natural k and n, n ≥ N(Y, k) = const, we construct a trigonometric polynomial T
n
of order ≤n that changes its monotonicity at the same points y
i
∈ Y as f and is such that
*20c || f - Tn || £ \fracc( k,s )n2\upomega k( f",1 \mathord\vphantom 1 n n ) ( || f - Tn || £ \fracc( r + k,s )nr\upomega k( f(r),1 \mathord | / |
\vphantom 1 n n ), f ? C(r), r 3 2 ), \begin{array}{*{20}{c}} {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {k,s} \right)}}{{{n^2}}}{{{\upomega }}_k}\left( {f',{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right)} \\ {\left( {\left\| {f - {T_n}} \right\| \leq \frac{{c\left( {r + k,s} \right)}}{{{n^r}}}{{{\upomega }}_k}\left( {{f^{(r)}},{1 \mathord{\left/{\vphantom {1 n}} \right.} n}} \right),\quad f \in {C^{(r)}},\quad r \geq 2} \right),} \\ \end{array} 相似文献
9.
Lou Yuanren 《分析论及其应用》1990,6(1):46-64
Let f∈Ap. For any positive integer l, the quantity Δ1,n−1(f:z) has been studied extensively. Here we give some quantitative estimates for
and investigate some pointwise estimates of Δ
l,n−1
(r)
(f;z).
Supported by National Science Foundation of China 相似文献
10.
J. J. Macys 《Lithuanian Mathematical Journal》2005,45(3):284-291
We consider the factorial quotients (2n − 1)!!/(2n)!! in connection with the Wallis formula n
−1(2n)!!2/(2n − 1)!!2 → π. We improve the Wallis inequalities (n + 1/2)−1(2n)!!2/(2n − 1)!!2 < π < n
−1(2n)!!2/(2n − 1)!!2 for π and obtain new estimates of factorial quotients with error order not worse than 1/n
2.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 349–358, July–September, 2005. 相似文献
11.
Jean-Christophe Bourgoin 《Calculus of Variations and Partial Differential Equations》2006,25(4):469-489
In this paper, we investigate the minimality of the map
from the Euclidean unit ball Bn to its boundary 핊n−1 for weighted energy functionals of the type Ep,f = ∫Bn f(r)‖∇ u‖p dx, where f is a non-negative function. We prove that in each of the two following cases:
12.
M. Zähle 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2009,44(2):117-145
Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brownian motion in ℝ
n
with generator f(−Δ) are investigated. Estimates for the related Rieszand Bessel-type kernels of order s are derived which include the classical
case f(r) = r
α/2 with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernstein functions f potential representations of the trace spaces, the trace Dirichlet forms, and the trace processes on fractal h-sets are derived. Here we suppose the trace condition ∫01
r
−(n+1)
f(r
−2)−1
h(r) dr < ∞ on f and the gauge function h.
Dedicated to the 80th birthday of Klaus Krickeberg 相似文献
13.
Let f ∈ L
w
1
[−1, 1], let r
n,m(f) be the best rational L
w
1
-approximation for f with respect to real rational functions of degree at most n in the numerator and of degree at most m in the denominator, let m = m(n), and let lim
n → ∞ (n-m(n)) = ∞. In this case, we show that the counting measures of certain subsets of sign changes of f-r
n,m
(f) converge weakly to the equilibrium measure on [−1, 1] as n → ∞. Moreover, we prove estimates for discrepancy between these counting measures and the equilibrium measure.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 283–287, February, 2006. 相似文献
14.
Raphael Yuster 《Graphs and Combinatorics》2001,17(3):579-587
We prove that for every ε>0 and positive integer r, there exists Δ0=Δ0(ε) such that if Δ>Δ0 and n>n(Δ,ε,r) then there exists a packing of K
n
with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most εn
2 edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let αΔ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥αΔ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K
n
).
Received: March 15, 1999?Final version received: October 22, 1999 相似文献
15.
Speed of convergence in nonparametric kernel estimation of a regression function and its derivatives
Alexander A. Georgiev 《Annals of the Institute of Statistical Mathematics》1984,36(1):455-462
Summary The objective in nonparametric regression is to infer a functiong(x) and itspth order derivativesg
(g)(x),p≧1 fixed, on the basis of a finite collection of pairs {x
i, g(xi)+Z
i}
i=1
n
, where the noise componentsZ
i satisfy certain modest assumptions and the domain pointsx
i are selected non-randomly. This paper exhibits a new class of kernel estimatesg
n
(p)
,p≧0 fixed. The main theoretical results of this study are the rates of convergence obtained for mean square and strong consistency
ofg
n
(p)
each of them being uniform on the (0,1). 相似文献
16.
For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β^-1≤n1/n2 ≤β is provided with some fixed parameter ~ 〉 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property. 相似文献
17.
We present some exponential inequalities for positively associated unbounded random variables. By these inequalities, we obtain
the rate of convergence n
−1/2
β
n
log 3/2
n in which β
n
can be particularly taken as (log log n)1/σ
with any σ>2 for the case of geometrically decreasing covariances, which is faster than the corresponding one n
−1/2(log log n)1/2log 2
n obtained by Xing, Yang, and Liu in J. Inequal. Appl., doi: (2008) for the case mentioned above, and derive the convergence rate n
−1/2
β
n
log 1/2
n for the above β
n
under the given covariance function, which improves the relevant one n
−1/2(log log n)1/2log n obtained by Yang and Chen in Sci. China, Ser. A 49(1), 78–85 (2006) for associated uniformly bounded random variables. In addition, some moment inequalities are given to prove the main results,
which extend and improve some known results. 相似文献
18.
Carlo Magagna 《Monatshefte für Mathematik》2008,153(1):59-81
For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A
k
− I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative
to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces
relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational
points of an ordinary elliptic curve E over a finite field with q
n
elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that
the orders of the groups of
rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity.
Author’s address: Dipartimento di Matematica e Informatica, Via Delle Scienze 206, 33100 Udine, Italy 相似文献
19.
Accuracy of several multidimensional refinable distributions 总被引:3,自引:0,他引:3
Carlos Cabrelli Chritopher Heil Ursula Molter 《Journal of Fourier Analysis and Applications》2000,6(5):483-502
Compactly supported distributions f1,..., fr on ℝd are fefinable if each fi is a finite linear combination of the rescaled and translated distributions fj(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σk∈Λ ck f(Ax−k), where Λ is a finite subset of Γ, the ck are r×r matrices, and f=(f1,...,fr)T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced
from linear combinations of translates of f1,...,fr along the lattice Γ. We determine the accuracy p from the matrices ck. Moreover, we determine explicitly the coefficients yα,i(k) such that xα=Σ
i=1
r
Σk∈Γyα,i(k) fi(x+k). These coefficients are multivariate polynomials yα,i(x) of degree |α| evaluated at lattice points k∈Γ. 相似文献
20.
We study numerical integration for functions f with singularities. Nonadaptive methods are inefficient in this case, and we show that the problem can be efficiently solved by adaptive quadratures at cost similar to that for functions with no singularities.
Consider first a class of functions whose derivatives of order up to r are continuous and uniformly bounded for any but one singular point. We propose adaptive quadratures Q*n, each using at most n function values, whose worst case errors are proportional to n−r. On the other hand, the worst case error of nonadaptive methods does not converge faster than n−1.
These worst case results do not extend to the case of functions with two or more singularities; however, adaption shows its
power even for such functions in the asymptotic setting. That is, let F∞r be the class of r-smooth functions with arbitrary (but finite) number of singularities. Then a generalization of Q*n yields adaptive quadratures Q**n such that |I(f)−Q**n(f)|=O(n−r) for any f ∈ F∞r. In addition, we show that for any sequence of nonadaptive methods there are `many' functions in F∞r for which the errors converge no faster than n−1.
Results of numerical experiments are also presented.
The authors were partially supported, respectively, by the State Committee for Scientific Research of Poland under Project
1 P03A 03928 and by the National Science Foundation under Grant CCR-0095709. 相似文献
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