共查询到20条相似文献,搜索用时 321 毫秒
1.
The following estimate of the pth derivative of a probability density function is examined: , where hk is the kth Hermite function and Σi = 1nhk(p)(Xi) is calculated from a sequence X1,…, Xn of independent random variables having the common unknown density. If the density has r derivatives the integrated square error converges to zero in the mean and almost completely as rapidly as O(n?α) and O(n?α log n), respectively, where . Rates for the uniform convergence both in the mean square and almost complete are also given. For any finite interval they are O(n?β) and , respectively, where . 相似文献
2.
Robert Chen 《Journal of multivariate analysis》1978,8(2):328-333
Let {Xn}n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let Sn = Σj=1nXj and . In this paper, we prove that (1) lim?→0+?α(r?1)E{N∞(r, t, ?)} = K(r, t) if E(X1) = 0, Var(X1) = 1, and E(| X1 |t) < ∞, where 2 ≤ t < 2r ≤ 2t, , and ; (2) if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N∞(t, t, ?)} = Σn=1∞nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and , i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(| X1 |t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution. 相似文献
3.
Tom M. Apostol 《Journal of Number Theory》1982,15(1):14-24
An elementary proof is given of the author's transformation formula for the Lambert series relating Gp(e2πiτ) to Gp(e2πiAτ), where p > 1 is an odd integer and is a general modular substitution. The method extends Sczech's argument for treating Dedekind's function , and uses Carlitz's formula expressing generalized Dedekind sums in terms of Eulerian functions. 相似文献
4.
Letting G(n) denote the number of nonisomorphic groups of order n, it is shown that for square-free n, G(n) ≤ ?(n) and G(n) ≤ (log n)c on a set of positive density. Letting Fk(x) denote the number of n ≤ x for which G(n) = k, it is shown that , where logrx denotes the r-fold iterated logarithm. 相似文献
5.
L.R. Haff 《Journal of multivariate analysis》1977,7(3):374-385
Let Sp×p ~ Wishart (Σ, k), Σ unknown, k > p + 1. Minimax estimators of Σ?1 are given for L1, an Empirical Bayes loss function; and L2, a standard loss function (Ri ≡ E(Li ∣ Σ), i = 1, 2). The estimators are , a, b ≥ 0, r(·) a functional on . Stein, Efron, and Morris studied the special cases and , for certain, a, b. From their work , a = k ? p ? 1, b = p2 + p ? 2; whereas, we prove . The reversal is surprising because a.e. (for a particular L2). Assume (compact) ? , the set of p × p p.s.d. matrices. A “divergence theorem” on functions Fp×p : → implies identities for Ri, i = 1, 2. Then, conditions are given for , i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p ∣S∣1/p/tr(S). 相似文献
6.
7.
Patrick J Browne 《Journal of Differential Equations》1977,23(2):285-292
In this paper we study the linked nonlinear multiparameter system , where xr? [ar, br], yr is subject to Sturm-Liouville boundary conditions, and the continuous functions ars satisfy ¦ . Conditions on the polynomial operators Mr, Prs are produced which guarantee a sequence of eigenfunctions for this problem yn(x) = Πr=1kyrn(xr), n ? 1, which form a basis in . Here [a, b] = [a1, b1 × … × [ak, bk]. 相似文献
8.
In two party elections with popular vote ratio , a theoretical model suggests replacing the so-called MacMahon cube law approximation , for the ratio of candidates elected, by the ratio of the two half sums in the binomial expansion of (p+q)2k+1 for some k. This ratio is nearly when k = 6. The success probability for the power law is shown to so closely approximate , if we choose , that for . Computationally, we avoid large binomial coefficients in computing for k>22 by expressing as the sum , whose terms decrease by the factors . Setting K = 4k+3, we compute ak for the large k using a continued fraction derived from the ratio of π to the finite Wallis product approximation. 相似文献
9.
J.E Nymann 《Journal of Number Theory》1975,7(4):406-412
Given a set S of positive integers let denote the number of k-tuples 〈m1, …, mk〉 for which and (m1, …, mk) = 1. Also let denote the probability that k integers, chosen at random from , are relatively prime. It is shown that if P = {p1, …, pr} is a finite set of primes and S = {m : (m, p1 … pr) = 1}, then if k ≥ 3 and where d(S) denotes the natural density of S. From this result it follows immediately that as n → ∞. This result generalizes an earlier result of the author's where and S is then the whole set of positive integers. It is also shown that if S = {p1x1 … prxr : xi = 0, 1, 2,…}, then as n → ∞. 相似文献
10.
Cai Mao-cheng 《Discrete Mathematics》1982,41(3):229-234
Let G be a minimally k-connected graph of order n and size e(G).Mader [4] proved that (i) ; (ii) e(G)?k(n?k) if n?3k?2, and the complete bipartite graph Kk,n?k is the only minimally k-connected graph of order; n and size k(n?k) when k?2 and n?3k?1.The purpose of the present paper is to determine all minimally k-connected graphs of low order and maximal size. For each n such that k+1?n?3k?2 we prove and characterize all minimally k-connected graphs of order n and size . 相似文献
11.
The problem of constructing a maximal t-linearly independent set in V(r; s) (called a maximal Lt(r, s)-set in this paper) is a very important one (called a packing problem) concerning a fractional factorial design and an error correcting code where V(r; s) is an r-dimensional vector space over a Galois field GF(s) and s is a prime or a prime power. But it is very difficult to solve it and attempts made by several research workers have been successful only in special cases.In this paper, we introduce the concept of a {Σα=1kwα, m; t, s}-min · hyper with weight (w1, w2,…, wk) and using this concept and the structure of a finite projective geometry PG(n ? 1, s), we shall give a geometrical method of constructing a maximal Lt(t + r, s)-set with length t + r + n for any integers r, n, and s such that n ? 3, n ? 1 ? r0 ? n + s ? 2 and r1 ? 1, where r = (r1 + 1)vn?1 ? r0 and . 相似文献
12.
The system is investigated, where x and y are scalar functions of time (t ? 0), and n space variables , and F and G are nonlinear functions. Under certain hypotheses on F and G it is proved that there exists a unique spherically symmetric solution , which is bounded for r ? 0 and satisfies x(0) >x0, y(0) > y0, x′(0) = 0, y′(0) = 0, and x′ < 0, y′ > 0, ?r > 0. Thus, (x(r), y(r)) represents a time independent equilibrium solution of the system. Further, the linearization of the system restricted to spherically symmetric solutions, around (x(r), y(r)), has a unique positive eigenvalue. This is in contrast to the case n = 1 (i.e., one space dimension) in which zero is an eigenvalue. The uniqueness of the positive eigenvalue is used in the proof that the spherically symmetric solution described is unique. 相似文献
13.
Patrick J. Browne 《Journal of Mathematical Analysis and Applications》1977,60(1):259-273
In this paper we continue our investigation of multiparameter spectral theory. Let H1,…, Hk be separable Hilbert spaces and H = ?r = 1kHr, be their tensor product. In each space Hr we have densely defined self-adjoint operators Tr and continuous Hermitian operators Vrs. The multiparameter eigenvalue problem concerns eigenvalues λ = (λ1,…, λn) ?Rk and eigenvectors such that . We develop a spectral theory for such systems leading to a Parseval equality and generalized eigenvector expansion. The results are applied to a k × k system of linked secondorder differential equations. 相似文献
14.
Rudolf Wegmann 《Journal of Mathematical Analysis and Applications》1976,56(1):113-132
For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by · number {λi: λi ? x} for all real x. Let An for n = 1, 2, … be an n × n matrix, whose entries aik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, , p) with the same distribution Fa. Suppose that all moments | a | k, k = 1, 2, … are finite, a=0 and | a | 2. Let with complex numbers θσ and finite products Pσ of factors A and (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G0(x) = G1x) + G2(x), where G1 is a step function and G2 is absolutely continuous, such that with probability converges to G0(x) as n → ∞ for all continuity points x of G0. The density g of G2 vanishes outside a finite interval. There are only finitely many jumps of G1. Both, G1 and G2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigner's semicircle theorem for symmetric random matrices (E. P. Wigner, Random matrices in physics, SIAM Review9 (1967), 1–23). The examples , , , r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form of Schur's inequality for the eigenvalues λi(n) of An holds. Consequently random matrices do not tend to be normal matrices for large n. 相似文献
15.
For a sequence A = {Ak} of finite subsets of N we introduce: , , where A(m) is the number of subsets Ak ? {1, 2, …, m}.The collection of all subsets of {1, …, n} together with the operation constitutes a finite semi-group N∪ (semi-group N∩) (group ). For N∪, N∩ we prove analogues of the Erdös-Landau theorem: δ(A+B) ? δ(A)(1+(2λ)?1(1?δ(A>))), where B is a base of N of the average order λ. We prove for analogues of Schnirelmann's theorem (that δ(A) + δ(B) > 1 implies δ(A + B) = 1) and the inequalities λ ? 2h, where h is the order of the base.We introduce the concept of divisibility of subsets: a|b if b is a continuation of a. We prove an analog of the Davenport-Erdös theorem: if d(A) > 0, then there exists an infinite sequence {Akr}, where Akr | Akr+1 for r = 1, 2, …. In Section 6 we consider for analogues of Rohrbach inequality: , where g(n) = min k over the subsets {a1 < … < ak} ? {0, 1, 2, …, n}, such that every m? {0, 1, 2, …, n} can be expressed as m = ai + aj.Pour une série A = {Ak} de sous-ensembles finis de N on introduit les densités: , où A(m) est le nombre d'ensembles Ak ? {1, 2, …, m}. L'ensemble de toutes les parties de {1, 2, …, n} devient, pour les opérations , un semi-groupe fini N∪, N∩ ou un groupe N1 respectivement. Pour N∪, N∩ on démontre l'analogue du théorème de Erdös-Landau: δ(A + B) ? δ(A)(1 + (2λ)?1(1?δ(A))), où B est une base de N d'ordre moyen λ. On démontre pour l'analogue du théorème de Schnirelmann (si δ(A) + δ(B) > 1, alors δ(A + B) = 1) et les inégalités λ ? 2h, où h est l'ordre de base. On introduit le rapport de divisibilité des enembles: a|b, si b est une continuation de a. On démontre l'analogue du théorème de Davenport-Erdös: si d(A) > 0, alors il existe une sous-série infinie {Akr}, où Akr|Akr+1, pour r = 1, 2, … . Dans le Paragraphe 6 on envisage pour N∪, les analogues de l'inégalité de Rohrbach: , où g(n) = min k pour les ensembles {a1 < … < ak} ? {0, 1, 2, …, n} tels que pour tout m? {0, 1, 2, …, n} on a m = ai + aj. 相似文献
16.
Let k and r be fixed integers such that 1 < r < k. Any positive integer n of the form n = akb, where b is r-free, is called a (k, r)-integer. In this paper we prove that if Qk,r(x) denotes the number of (k, r)-integers ≤ x, then , where , B being a positive constant depending on r and the O-estimate is uniform in k. On the assumption of the Riemann hypothesis, we improve the above order estimate of Δk,r(x) and prove that , according as or , where ω(x) = exp [B log x(log log x)?1]. 相似文献
17.
Given an integer k>0, our main result states that the sequence of orders of the groups (respectively, of the groups ) is Cesàro equivalent as n→∞ to the sequence C1(k)nk2?1 (respectively, C2(k)nk2), where the coefficients C1(k) and C2(k) depend only on k; we give explicit formulas for C1(k) and C2(k). This result generalizes the theorem (which was first published by I. Schoenberg) that says that the Euler function ?(n) is Cesàro equivalent to . We present some experimental facts related to the main result. To cite this article: A.G. Gorinov, S.V. Shadchin, C. R. Acad. Sci. Paris, Ser. I 337 (2003). 相似文献
18.
C.G Khatri 《Journal of multivariate analysis》1978,8(3):453-467
Optimization problems are connected with maximization of three functions, namely, geometric mean, arithmetic mean and harmonic mean of the eigenvalues of (X′ΣX)?1ΣY(Y′ΣY)?1Y′ΣX, where Σ is positive definite, X and Y are p × r and p × s matrices of ranks r and s (≥r), respectively, and X′Y = 0. Some interpretations of these functions are given. It is shown that the maximum values of these functions are obtained at the same point given by X = (h1 + ?1hp, …, hr + ?rhp?r+1) and , where h1, …, hp are the eigenvectors of Σ corresponding to the eigenvalues λ1 ≥ λ2 ≥ … ≥ λp > 0, ?j = +1 or ?1 for j = 1,2,…, r and , are linear functions of hr+1,…, hp?r. These results are extended to intermediate stationary values. They are utilized in obtaining the inequalities for canonical correlations θ1,…,θr and they are given by expressions (3.8)–(3.10). Further, some new union-intersection test procedures for testing the sphericity hypothesis are given through test statistics (3.11)–(3.13). 相似文献
19.
Allen J. Schwenk 《Discrete Mathematics》1977,18(1):71-78
Let denote the polynomial obtained from the cycle index of the symmetric group Z(Sn) by replacing each variable si by f(x1). Let f(x) have a Taylor series with radius of convergence ? of the form f(x)=xk + ak+1xk+1 + ak+2xk+2+? with every a1?0. Finally, let 0<x<1 and let x??. We prove that This limit is used to estimate the probability (for n and p both large) that a point chosen at random from a random p-point tree has degree n + 1. These limiting probabilities are independent of p and decrease geometrically in n, contrasting with the labeled limiting probabilities of .In order to prove the main theorem, an appealing generalization of the principle of inclusion and exclusion is presented. 相似文献
20.
Shlomo Moran 《Journal of Combinatorial Theory, Series B》1984,37(2):113-141
Let V be a set of n points in Rk. Let d(V) denote the diameter of V, and l(V) denote the length of the shortest circuit which passes through all the points of V. (Such a circuit is an “optimal TSP circuit”.) lk(n) are the extremal values of l(V) defined by lk(n)=max{l(V)|V∈Vnk}, where Vnk={V|V?Rk,|V|=n, d(V)=1}. A set V∈Vnk is “longest” if l(V)=lk(n). In this paper, first some geometrical properties of longest sets in R2 are studied which are used to obtain l2(n) for small n′s, and then asymptotic bounds on lk(n) are derived. Let δ(V) denote the minimal distance between a pair of points in V, and let: δk(n)=max{δ(V)|V∈Vnk}. It is easily observed that . Hence, exists. It is shown that for all , and hence, for all . For k=2, this implies that , which generalizes an observation of Fejes-Toth that . It is also shown that . The above upper bound is used to improve related results on longest sets in k-dimensional unit cubes obtained by Few (Mathematika2 (1955), 141–144) for almost all k′s. For k=2, Few's technique is used to show that . 相似文献