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1.
Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 +... + Xn, n≥1. The author proves that, if EX^2I{|X|≥t} = 0((log log t)^-1) as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1+a(1/√1+a-ε)b+1 ∑n=1^∞(logn)^a(loglogn)^b/nP{max κ≤n|Sκ|≤√σ^2π^2n/8loglogn(ε+an)}=4/π(1/2(1+a)^3/2)^b+1 Г(b+1),whenever an = o(1/log log n). The author obtains the sufficient and necessary conditions for this kind of results to hold.  相似文献   

2.
Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn).  相似文献   

3.
LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetGL 1 (X, Σ,μ) bem-dimensional. Theorem:There is an l ∈ E* and real numbers −∞=x 0<x 1<x 2<…<x n<x n+1=∞with nm, such that for all g ∈ G,   相似文献   

4.
Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1+X2+…+Xn,Mn=max k≤n│Sk│,n≥1.Let an=O(1/loglogn).In this paper,we prove that,for b〉-1,lim ε→0 →^2(b+1)∑n=1^∞ (loglogn)^b/nlogn n^1/2 E{Mn-σ(ε+an)√2nloglogn}+σ2^-b/(b+1)(2b+3)E│N│^2b+3∑k=0^∞ (-1)k/(2k+1)^2b+3 holds if and only if EX=0 and EX^2=σ^2〈∞.  相似文献   

5.
By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation given by ( a1, …, an). Assume that either a1, …, an are positive rational numbers or for some Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ℝn must be infinite  相似文献   

6.
We prove a general theorem on the zeros of a class of generalised Dirichlet series. We quote the following results as samples. Theorem A.Let 0<θ<1/2and let {a n }be a sequence of complex numbers satisfying the inequality for N = 1,2,3,…,also for n = 1,2,3,…let α n be real andn| ≤ C(θ)where C(θ) > 0is a certain (small)constant depending only on θ. Then the number of zeros of the function in the rectangle (1/2-δ⩽σ⩽1/2+δ,Tt⩽2T) (where 0<δ<1/2)isC(θ,δ)T logT where C(θ,δ)is a positive constant independent of T provided TT 0(θ,δ)a large positive constant. Theorem B.In the above theorem we can relax the condition on a n to and |aN| ≤ (1/2-θ)-1.Then the lower bound for the number of zeros in (σ⩾1/3−δ,Tt⩽2T)is > C(θ,δ) Tlog T(log logT)-1.The upper bound for the number of zeros in σ⩾1/3+δ,Tt⩽2T) isO(T)provided for every ε > 0. Dedicated to the memory of Professor K G Ramanathan  相似文献   

7.
In the middle of the 20th century Hardy obtained a condition which must be imposed on a formal power series f(x) with positive coefficients in order that the series f −1(x) = $ \sum\limits_{n = 0}^\infty {b_n x^n } $ \sum\limits_{n = 0}^\infty {b_n x^n } b n x n be such that b 0 > 0 and b n ≤ 0, n ≥ 1. In this paper we find conditions which must be imposed on a multidimensional series f(x 1, x 2, …, x m ) with positive coefficients in order that the series f −1(x 1, x 2, …, x m ) = $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } $ \sum i_1 ,i_2 , \ldots ,i_m \geqslant 0^b i_1 ,i_2 , \ldots ,i_m ^{x_1^{i_1 } x_2^{i_2 } \ldots x_m^{i_m } } satisfies the property b 0, …, 0 > 0, $ bi_1 ,i_2 , \ldots ,i_m $ bi_1 ,i_2 , \ldots ,i_m ≤ 0, i 12 + i 22 + … + i m 2 > 0, which is similar to the one-dimensional case.  相似文献   

8.
Let (X, Xn; n ≥1) be a sequence of i.i.d, random variables taking values in a real separable Hilbert space (H, ||·||) with covariance operator ∑. Set Sn = X1 + X2 + ... + Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2(b+1) ∞ ∑n=1 (logn)^b/n^3/2 E{||Sn||-σε√nlogn}=σ^-2(b+1)/(2b+3)(b+1) B||Y|^2b+3
holds if EX=0,and E||X||^2(log||x||)^3bv(b+4)〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.  相似文献   

9.
If N ∈ ℕ, 0 < p ≤ 1, and(Xk) k=1 N are r.i.p-spaces, it is shown that there is C(= C(p, N)) > 0, such that for every ƒ ∈ ∩ k=1 N Xk, there exists with , for every 1 ≤ k ≤ N. Also, if ⊓ is a convex polygon in ℝ2, it is proved that the N-tuple (H(X1),…, H(Xn)) is K-closed with respect to (X1,…, XN) in the sense of Pisier. Everything follows from Theorem 2.1, which is a general analytic partition of unity type result.  相似文献   

10.
In this paper, we investigate the complex oscillation of the differential equation
whereA k−1, …,A 0, F # 0 are finite order transcendental entire functions, such that there exists anA d(0≤d≤k−1) being dominant in the sense that either it has larger order than any otherA j(j=0.…,d−1, d+1.…, k−1), or it is the only transcendental function We obtain some precise estimates of the exponent of convergence of the zero-sequence of solutions to the above equation. Project supported by the National Natural Science Foundation of China  相似文献   

11.
Let G n,k be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc n12,…, β k ), β12,…, βk = 0,1,2,…, β12 + … +β k n,c n(0,0,…, 0) = 1 and whenever β0n - (β1 + β2 + … + β k ) where Δc n12,…, β k ) =c n1 + 1, β2,…, β k )+c n12+1,…, β k )+…+c n12,…, β k +1) -c n12,…, β k ). Further, let Π n,k be the set of all symmetric probabilities on {0,1,2,…,k} n . We establish a one-to-one correspondence between the sets G n,k and Π n,k and use it to formulate and answer interesting questions about both. Assigning to G n,k the uniform probability measure, we show that, asn→∞, any fixed section {it{cn}(β12,…, β k ), 1 ≤ Σβ i m}, properly centered and normalized, is asymptotically multivariate normal. That is, converges weakly to MVN[0, Σ m ]; the centering constantsc 01, β2,…, β k ) and the asymptotic covariances depend on the moments of the Dirichlet (1, 1,…, 1; 1) distribution on the standard simplex inR k.  相似文献   

12.
Statistical properties of continued fractions for numbers a/b, where a and b lie in the sector a, b ≥ 1, a2 + b2 ≤ R2, are studied. The main result is an asymptotic formula with two meaning terms for the quantity
where sx(a/b) = |{j ε {1, …, s}: [0; tj, …, ts] ≤ x}| is the Gaussian statistic for the fraction a/b = [t0; t1, …, ts]. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 186–211.  相似文献   

13.
A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, yX. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (xn) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (xn) be aT-martingale taking on values inH. If then x n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (xn) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X * of norm 1 such that If, in addition, the spaceX is strictly convex, x n /n converges weakly; and if the norm ofX * is Fréchet differentiable (away from zero), x n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093  相似文献   

14.
Let be a measure on R and let σ = (m 1, m 2,...,m n ), where m k ≥ 1, k = 1,2,...,n, are arbitrary real numbers. A polynomial ω n (x) := (xx 1)(xx 2)...(xx n ) with x 1x 2 ≤ ... ≤ x n is said to be the nth σ-orthogonal polynomial with respect to if the vector of zeros (x 1, x 2, ..., x n)T is a solution of the extremal problem
In this paper the existence, uniqueness, characterizations, and continuity with respect to σ of a σ-orthogonal polynomial under a more mild assumption are established. An efficient iterative method based on solving the system of normal equations for constructing a σ-orthogonal polynomial is presented when all the m k are arbitrary real numbers no less than 2. A simple method to calculate the Cotes numbers of the corresponding generalized Gaussian quadrature formula when all the m k are positive integers no less than 2 is provided. Finally, some numerical examples are also given. Support in part by Natural Science Foundation of China under grants 10241004 and 10371130.  相似文献   

15.
LetK be a field, charK=0 andM n (K) the algebra ofn×n matrices overK. If λ=(λ1,…,λ m ) andμ=(μ 1,…,μ m ) are partitions ofn 2 let wherex 1,…,x n 2,y 1,…,y n 2 are noncommuting indeterminates andS n 2 is the symmetric group of degreen 2. The polynomialsF λ, μ , when evaluated inM n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF λ, μ is a central polynomial (not a polynomial identity) forM n (K). We give a formula that allows us to evaluateF λ, μ inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF λ, μ is a polynomial identity forM n (K). As an application, we exhibit a new class of central polynomials forM n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.  相似文献   

16.
Let {Xk} be a sequence of i.i.d. random variables with d.f. F(x). In the first part of the paper the weak convergence of the d.f.'s Fn(x) of sums is studied, where 0<α≤2, ank>0, 1≤k≤mn, and, as n→∞, bothmax 1≤k≤mna nk→0 and . It is shown that such convergence, with suitably chosen An's and necessarily stable limit laws, holds for all such arrays {αnk} provided it holds for the special case αnk=1/n, 1≤k≤n. Necessary and sufficient conditions for such convergence are classical. Conditions are given for the convergence of the moments of the sequence {Fn(x)}, as well as for its convergence in mean. The second part of the paper deals with the almost sure convergence of sums , where an≠0, bn>0, andmax 1≤k≤n ak/bn→0. The strong law is said to hold if there are constants An for which Sn→0 almost surely. Let N(0)=0 and N(x) equal the number of n≥1 for which bn/|an|<x if x>0. The main result is as follows. If the strong law holds,EN (|X1|)<∞. If for some 0<p≤2, then the strong law holds with if 1≤p≤2 and An=0 if 0<p<1. This extends the results of Heyde and of Jamison, Orey, and Pruitt. The strong law is shown to hold under various conditions imposed on F(x), the coefficients an and bn, and the function N(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.  相似文献   

17.
Summary In the paper we estimate a regressionm(x)=E {Y|X=x} from a sequence of independent observations (X 1,Y 1),…, (X n, Yn) of a pair (X, Y) of random variables. We examine an estimate of a type , whereN depends onn andϕ N is Dirichlet kernel and the kernel associated with the hermite series. Assuming, that E|Y|<∞ and |Y|≦γ≦∞, we give condition for to converge tom(x) at almost allx, provided thatX has a density. if the regression hass derivatives, then converges tom(x) as rapidly asO(nC−(2s−1)/4s) in probability andO(n −(2s−1)/4s logn) almost completely.  相似文献   

18.
We consider a (possibly) vector-valued function u: Ω→R N, Ω⊂R n, minimizing the integral , whereD iu=∂u/∂x i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D 1u,…,Dn−1u∈Lq, under suitable assumptions ona i(x).
Sunto Consideriamo una funzione u: Ω→R N, Ω⊂R n che minimizzi l'integrale , doveD iu=∂u/∂xi, o un funzionale con un comportamento simile; sotto opportune ipotesi sua i(x), dimostriamo la seguente maggiore integrabilità perDu:D 1u,…,Dn−1uεLq.
  相似文献   

19.
The equations under consideration have the following structure:
where 0 < x n < ∞, (x 1, …, x n−1) ∈ Ω, Ω is a bounded Lipschitz domain, is a function that is continuous and monotonic with respect to u, and all coefficients are bounded measurable functions. Asymptotic formulas are established for solutions of such equations as x n → + ∞; the solutions are assumed to satisfy zero Dirichlet or Neumann boundary conditions on ∂Ω. Previously, such formulas were obtained in the case of a ij, ai depending only on (x 1, …, x n−1). __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 98–111, 2005.  相似文献   

20.
Let N+2m ={−m, −m+1, …, −1, 0, 1, …,N−1,N, …,N−1+m}. The present paper is devoted to the approximation of discrete functions of the formf : N+2m → ℝ by algebraic polynomials on the grid Ω N ={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets Ω N+m and Ω N , respectively, we construct a linear operatorY n+2m, N =Y n+2m, N (f), acting in the space of discrete functions as an algebraic polynomial of degree at mostn+2m for which the following estimate holds (x ε Ω N ):
(1)
whereE n+m[g,l 2 N+m )] is the best approximation of the function
(1)
by algebraic polynomials of degree at mostn+m in the spacel 2 N+m ) and the function Θ N, α (x) depends only on the weighted estimate for the Chebyshev polynomialsτ k α,α (x, N). Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 460–470, March, 2000.  相似文献   

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