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1.
The following estimate of the pth derivative of a probability density function is examined: Σk = 0Na?khk(x), where hk is the kth Hermite function and a?k = ((?1)pn)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 α = 2(r ? p)(2r + 1). 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 O(n2log n), respectively, where β = (2(r ? p) ? 1)(2r + 1).  相似文献   

2.
Let X1, X2, X3, … be i.i.d. r.v. with E|X1| < ∞, E X1 = μ. Given a realization X = (X1,X2,…) and integers n and m, construct Yn,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution P1(Yn,i = Xj) = 1n for 1 ? j ? n. (P1 denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X1 are given to ensure the almost sure convergence of (1mmi=1 Yn,i toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).  相似文献   

3.
This paper is a study of the distribution of eigenvalues of various classes of operators. In Section 1 we prove that the eigenvalues (λn(T)) of a p-absolutely summing operator, p ? 2, satisfy
n∈Nn(T)|p1pp(T).
This solves a problem of A. Pietsch. We give applications of this to integral operators in Lp-spaces, weakly singular operators, and matrix inequalities.In Section 2 we introduce the quasinormed ideal Π2(n), P = (p1, …, pn) and show that for TΠ2(n), 2 = (2, …, 2) ∈ Nn, the eigenvalues of T satisfy
i∈Ni(T)|2nn2n2(T).
More generally, we show that for TΠp(n), P = (p1, …, pn), pi ? 2, the eigenvalues are absolutely p-summable,
1p=i=1n1piandn∈Nn(T)|p1p?CpπnP(T).
We also consider the distribution of eigenvalues of p-nuclear operators on Lr-spaces.In Section 3 we prove the Banach space analog of the classical Weyl inequality, namely
n∈Nn(T)|p ? Cpn∈N αn(T)p
, 0 < p < ∞, where αn denotes the Kolmogoroff, Gelfand of approximation numbers of the operator T. This solves a problem of Markus-Macaev.Finally we prove that Hilbert space is (isomorphically) the only Banach space X with the property that nuclear operators on X have absolutely summable eigenvalues. Using this result we show that if the nuclear operators on X are of type l1 then X must be a Hilbert space.  相似文献   

4.
Let Ω be a simply connected domain in the complex plane, and A(Ωn), the space of functions which are defined and analytic on Ωn, if K is the operator on elements u(t, a1, …, an) of A(Ωn + 1) defined in terms of the kernels ki(t, s, a1, …, an) in A(Ωn + 2) by Ku = ∑i = 1naitk i(t, s, a1, …, an) u(s, a1, …, an) ds ? A(Ωn + 1) and I is the identity operator on A(Ωn + 1), then the operator I ? K may be factored in the form (I ? K)(M ? W) = (I ? ΠK)(M ? ΠW). Here, W is an operator on A(Ωn + 1) defined in terms of a kernel w(t, s, a1, …, an) in A(Ωn + 2) by Wu = ∝antw(t, s, a1, …, an) u(s, a1, …, an) ds. ΠW is the operator; ΠWu = ∝an ? 1w(t, s, a1, …, an) u(s, a1, …, an) ds. ΠK is the operator; ΠKu = ∑i = 1n ? 1aitki(t, s, a1, …, an) ds + ∝an ? 1tkn(t, s, a1, …, an) u(s, a1, …, an) ds. The operator M is of the form m(t, a1, …, an)I, where m ? A(Ωn + 1) and maps elements of A(Ωn + 1) into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator K1 on functions u in A(Ωn + 2), by K1u = ∑i = 1n ? 1ait ki(t, s, a1, …, an) u(s, a, …, an + 1) ds + ∝an + 1t kn(t, s, a1, …, an) u((s, a1, …, an + 1) ds. A determinant δ(I ? K1) of the operator I ? K1 is defined as an element m1(t, a1, …, an + 1) of A(Ωn + 2). This is mapped into A(Ωn + 1) by setting an + 1 = t to give m(t, a1, …, an). The operator I ? ΠK may be factored in similar fashion, giving rise to a chain factorization of I ? K. In some cases all the matrix kernels ki defining K are separable in the sense that ki(t, s, a1, …, an) = Pi(t, a1, …, an) Qi(s, a1, …, an), where Pi is a 1 × pi matrix and Qi is a pi × 1 matrix, each with elements in A(Ωn + 1), explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case.  相似文献   

5.
We consider the mixed boundary value problem Au = f in Ω, B0u = g0in Γ?, B1u = g1in Γ+, where Ω is a bounded open subset of Rn whose boundary Γ is divided into disjoint open subsets Γ+ and Γ? by an (n ? 2)-dimensional manifold ω in Γ. We assume A is a properly elliptic second order partial differential operator on Ω and Bj, for j = 0, 1, is a normal jth order boundary operator satisfying the complementing condition with respect to A on Γ+. The coefficients of the operators and Γ+, Γ? and ω are all assumed arbitrarily smooth. As announced in [Bull. Amer. Math. Soc.83 (1977), 391–393] we obtain necessary and sufficient conditions in terms of the coefficients of the operators for the mixed boundary value problem to be well posed in Sobolev spaces. In fact, we construct an open subset T of the reals such that, if Ds = {u ? Hs(Ω): Au = 0} then for s ? = 12(mod 1), (B0,B1): Ds → Hs ? 12?) × Hs ? 32+) is a Fredholm operator if and only if s ∈T . Moreover, T = ?xewTx, where the sets Tx are determined algebraically by the coefficients of the operators at x. If n = 2, Tx is the set of all reals not congruent (modulo 1) to some exceptional value; if n = 3, Tx is either an open interval of length 1 or is empty; and finally, if n ? 4, Tx is an open interval of length 1.  相似文献   

6.
In this paper we study trace formulas for a class of operators of the form ΠT-GΠT in which G designates multiplication by a suitable restricted d × d matrix valued function G(γ) of γ?R1and ΠT stands for the diagonal d × d matrix (δijPT) of orthogonal projections PT of L2(R1, ) onto the space IT() of entire functions of exponential type ?T which are square summable on the line relative to the measure dδ(γ) = ¦h(γ)¦2. It is shown that, for a reasonably large class of h,
limT↑∞trace[(?TG?T)n ? ?TGn?T]
exists and is independent of the choice of h within the permitted class. These results are then used to study the asymptotic behavior, as T ↑ ∞, of the determinant of I ? ΠTGΠT.  相似文献   

7.
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 (RiE(LiΣ), i = 1, 2). The estimators are Σ??1 = aS?1 + br(S)Ip×p, a, b ≥ 0, r(·) a functional on Rp(p+2)2. Stein, Efron, and Morris studied the special cases Σa?1 = aS?1 (EΣ?k?p?1?1 = Σ?1) and Σ?1?1 = aS?1 + (b/tr S)I, for certain, a, b. From their work R1?1, Σ?1?1; S) ≤ R1?1, Σ?a?1; S) (?Σ), a = k ? p ? 1, b = p2 + p ? 2; whereas, we prove R2?1Σ?a?1; S) ≤ R2?1, Σ?1?1; S) (?Σ). The reversal is surprising because L1?1, Σ?1?1; S) → L2?1, Σ?1?1; S) a.e. (for a particular L2). Assume R (compact) ? S, S the set of p × p p.s.d. matrices. A “divergence theorem” on functions Fp×p : RS implies identities for Ri, i = 1, 2. Then, conditions are given for Ri?1, Σ??1; S) ≤ Ri?1, Σ?1?1; S) ≤ Ri?1, Σ?a?1; S) (?Σ), i = 1, 2. Most of our results concern estimators with r(S) = t(U)/tr(S), U = p ∣S1/p/tr(S).  相似文献   

8.
Let θ(n) denote the maximum likelihood estimator of a vector parameter, based on an i.i.d. sample of size n. The class of estimators θ(n) + n?1q(θ(n)), with q running through a class of sufficiently smooth functions, is essentially complete in the following sense: For any estimator T(n) there exists q such that the risk of θ(n) + n?1q(θ(n)) exceeds the risk of T(n) by an amount of order o(n?1) at most, simultaneously for all loss functions which are bounded, symmetric, and neg-unimodal. If q1 is chosen such that θ(n) + n?1 q1(n)) is unbiased up to o(n?12), then this estimator minimizes the risk up to an amount of order o(n?1) in the class of all estimators which are unbiased up to o(n?12).The results are obtained under the assumption that T(n) admits a stochastic expansion, and that either the distributions have—roughly speaking—densities with respect to the lebesgue measure, or the loss functions are sufficiently smooth.  相似文献   

9.
Let m and vt, 0 ? t ? 2π be measures on T = [0, 2π] with m smooth. Consider the direct integral H = ⊕L2(vt) dm(t) and the operator (L?)(t, λ) = e?iλ?(t, λ) ? 2e?iλtT ?(s, x) e(s, t) dvs(x) dm(s) on H, where e(s, t) = exp ∫stTdvλ(θ) dm(λ). Let μt be the measure defined by T?(x) dμt(x) = ∫0tT ?(x) dvs dm(s) for all continuous ?, and let ?t(z) = exp[?∫ (e + z)(e ? z)?1t(gq)]. Call {vt} regular iff for all t, ¦?t(e)¦ = ¦?(e for 1 a.e.  相似文献   

10.
If r, k are positive integers, then Tkr(n) denotes the number of k-tuples of positive integers (x1, x2, …, xk) with 1 ≤ xin and (x1, x2, …, xk)r = 1. An explicit formula for Tkr(n) is derived and it is shown that limn→∞Tkr(n)nk = 1ζ(rk).If S = {p1, p2, …, pa} is a finite set of primes, then 〈S〉 = {p1a1p2a2psas; piS and ai ≥ 0 for all i} and Tkr(S, n) denotes the number of k-tuples (x1, x3, …, xk) with 1 ≤ xin and (x1, x2, …, xk)r ∈ 〈S〉. Asymptotic formulas for Tkr(S, n) are derived and it is shown that limn→∞Tkr(S, n)nk = (p1 … pa)rkζ(rk)(p1rk ? 1) … (psrk ? 1).  相似文献   

11.
The polynomial functions f1, f2,…, fm are found to have highest common factor h for a set of values of the variables x1, x2,…,xm whose asymptotic density is
1hnd∣hμ(d)Πml = 1 ?(f1, dh)dmΠp∣h1?Πml = 1?(f1, p)pm
For the special case f1(x) = f2(x) = … = fm(x) = x and h = 1 the above formula reduces to Π?(1 ? 1pm) = 1ζ(m), the density if m-tuples with highest common factor 1. Necessary and sufficient conditions on the polynomials f1, f2,…, fm for the asymptotic density to be zero are found. In particular it is shown that either the polynomials may never have highest common factor h or else h is the highest common factor infinitely often and in fact with positive density.  相似文献   

12.
In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z1…,zn), zi ? C, i=1, …, n, is uniquely determined from the magnitude of f(x1…,xn): | f(x1…,xn)|, xi ? R, i=1,…, n, except for (1) linear shifts: i(α1z1+…+αn2n+β), β, αi?R, i=1,…, n; and (2) conjugation: f1(z11,…,zn1).  相似文献   

13.
Let (A, G, α) be a C1-dynamical system, where G is abelian, and let φ be an invariant state. Suppose that there is a neighbourhood Ω of the identity in G? and a finite constant κ such that Πi = 1n φ(xi1xi) ? κ Πi = 1n φ(xixi1) whenever xi lies in a spectral subspace Rαi), where Ω1 + … + Ωn ? Ω. This condition of complete spectral passivity, together with self-adjointness of the left kernel of φ, ensures that φ satisfies the KMS condition for some one-parameter subgroup of G.  相似文献   

14.
Let V denote a finite dimensional vector space over a field K of characteristic 0, let Tn(V) denote the vector space whose elements are the K-valued n-linear functions on V, and let Sn(V) denote the subspace of Tn(V) whose members are the fully symmetric members of Tn(V). If Ln denotes the symmetric group on {1,2,…,n} then we define the projection PL : Tn(V) → Sn(V) by the formula (n!)?1Σσ ? Ln Pσ, where Pσ : Tn(V) → Tn(V) is defined so that Pσ(A)(y1,y2,…,yn = A(yσ(1),yσ(2),…,yσ(n)) for each A?Tn(V) and yi?V, 1 ? i ? n. If xi ? V1, 1 ? i ? n, then x1?x2? … ?xn denotes the member of Tn(V) such that (x1?x2· ? ? ?xn)(y1,y2,…,yn) = Пni=1xi(yi) for each y1 ,2,…,yn in V, and x1·x2xn denotes PL(x1?x2? … ?xn). If B? Sn(V) and there exists x i ? V1, 1 ? i ? n, such that B = x1·x2xn, then B is said to be decomposable. We present two sets of necessary and sufficient conditions for a member B of Sn(V) to be decomposable. One of these sets is valid for an arbitrary field of characteristic zero, while the other requires that K = R or C.  相似文献   

15.
For any tournament T on n vertices, let h(T) denote the maximum number of edges in the intersection of T with a transitive tournament on the same vertex set. Sharpening a previous result of Spencer, it is proved that, if Tn denotes the random tournament on n vertices, then, P(h(Tn) ≤ 12(2n) + 1.73n32) → 1 as n → ∞.  相似文献   

16.
Let (X, A) be a measurable space, Θ ? R an open interval and PΩA, Ω ? Θ, a family of probability measures fulfilling certain regularity conditions. Let Ωn be the maximum likelihood estimate for the sample size n. Let λ be a prior distribution on Θ and let Rn,x be the posterior distribution for the sample size n given x ? Xn. L: Θ × Θ → R denotes a loss function fulfilling certain regularity conditions and Tn denotes the Bayes estimate relative to λ and L for the sample size n. It is proved that for every compact K ? Θ there exists cK ≥ 0 such that
suptheta;∈KPtheta;nh{x∈Xn∥ Tn(x) ? ?nx|? cK(log n)n?} = o(n?12).
This theorem improves results of Bickel and Yahav [3], and Ibragimov and Has'minskii [4], as far as the speed of convergence is concerned.  相似文献   

17.
Let Fn(x) be the empirical distribution function based on n independent random variables X1,…,Xn from a common distribution function F(x), and let X = Σi=1nXin be the sample mean. We derive the rate of convergence of Fn(X) to normality (for the regular as well as nonregular cases), a law of iterated logarithm, and an invariance principle for Fn(X).  相似文献   

18.
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 E{N(r, t, ?)} = Σn=1 nr?2P{|Sn| > ?nrt}. 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, K(r, t) = {2α(r?1)2Γ((1 + α(r ? 1))2)}{(r ? 1) Γ(12)}, and α = 2t(2r ? t); (2) lim?→0+G(t, ?)H(t, ?) = 0 if 2 < t < 4, E(X1) = 0, Var(X1) > 0, and E(|X1|t) < ∞, where G(t, ?) = E{N(t, t, ?)} = Σn=1nt?2P{| Sn | > ?n} → ∞ as ? → 0+ and H(t, ?) = E{N(t, t, ?)} = Σn=1 nt?2P{| Sn | > ?n2t} → ∞ as ? → 0+, 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   

19.
Let Σ be an n × n positive definite matrix with eigenvalues λ1λ2 ≥ … ≥ λn > 0 and let M = {x, y | x?Rn, y?Rn, x ≠ 0, y ≠ 0, xy = 0}. Then for x, y in M, we have that x′Σy(x′Σxy′Σy)121 ? λn)1 + λn) and the inequality is sharp. If
∑=11122122
is a partitioning of Σ, let θ1 be the largest canonical correlation coefficient. The above result yields θ11 ? λn)1 + λn).  相似文献   

20.
Let PT denote the orthogonal projection of L2(R1, ) onto the space of entire functions of exponential type ? T which are square summable on the line with respect to the measure dΔ(γ) = ¦ h(γ)¦2, and let G denote the operator of multiplication by a suitably restricted complex valued function g. It is shown that if 2 + 1)?1log ¦ h(γ)¦ is summable, if ¦ h ¦?2 is locally summable, and if hh# belongs to the span in L of e?iyTH:T ? 0, in which h is chosen to be an outer function and h#(γ) agrees with the complex conjugate of h(γ) on the line, then
lim traceT↑∞{(PTGPT)n ? PTGnPT}
exists and is independent of h for every positive integer n. This extends the range of validity of a formula due to Mark Kac who evaluated this limit in the special case h = 1 using a different formalism. It also extends earlier results of the author which were established under more stringent conditions on h. The conclusions are based in part upon a preliminary study of a more general class of projections.  相似文献   

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