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1.
§ 1  Introduction and main resultsL et { X,Xn;n≥ 1} be a sequence of random variables with common distributionfunction F,mean0 and positive,finite variance,and set Sn= nk=1 Xk,n≥ 1.Also letlogx= ln(x∨e) ,log logx=log(logx) and(x) =2 xlog logx.Gut and Sp taru[2 ] studied theprecise asymptotics on the law of the iterated logarithm.One of their results is as follows.Theorem A.Spuuose that{ X ,Xn;n≥ 1} is a sequence of i.i.d.random variables with EX= 0 and0 相似文献   

2.
Let {X,X_n,n≥ 1} be a sequence of identically distributed pairwise negative quadrant dependent(PNQD) random variables and {a_n,n≥ 1} be a sequence of positive constants with a_n=f(n) and f(θ~k)/f(θ~(k-1)≥β for all large positive integers k, where 1 θ≤β and f(x) 0(x≥1) is a non-decreasing function on [b,+∞) for some b≥1.In this paper,we obtain the strong law of large numbers and complete convergence for the sequence {X,X_n, n≥ 1},which are equivalent to the general moment condition∑_(n=1)~∞ P(|X| a_n) ∞.Our results extend and improve the related known works in Baum and Katz [1],Chen at al.[3],and Sung[14].  相似文献   

3.
Let the time series {X(t),t=1,2,…}satisfy φ(B)(1-B)~dX(t)=θ(B)e(t),where B is a backward shift operator,defined by BX(t)=X(t-1),and φ(z)=1+φz+…+φ_pz~p,θ(z)=1+θ_1z+…+θ_qz~q%,and all the roots of φ(z)lie outside the unit circle;{e(t)}is a sequence of iid random variables with mean zero and E|e(t)|~(4+r)<∞(r>0).In this paper,the limit properties of S_n=sum from t=1 X(t)~2/t~(2d)log n,where the integer d≥1,have been considered.  相似文献   

4.
Let {X, X_n; n ≥ 0} be a sequence of independent and identically distributed random variables with EX=0, and assume that EX~2I(|X| ≤ x) is slowly varying as x →∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series Σ~∞_(n=0)β~nX_n(0 β 1) is obtained, under some minimal conditions.  相似文献   

5.
Let {X, X n , n≥1} be a sequence of i.i.d.random variables with zero mean, and set Sn = Σ k=1 n X k , EX2=σ 2>0, λ(ε) =Σ n=1 ∞ P (|Sn|≥ nε). In this paper, we discuss the rate of the approximation of σ2 by ε2 λ(ε) under suitable conditions, and improve the corresponding results of Klesov (1994).  相似文献   

6.
In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.  相似文献   

7.
Let f(x)∈L_(2π) and its Fourier series by f(x)~α_0/2+sum from n=1 to ∞(α_ncosnx+b_nsinx)≡sum from n=0 to ∞(A_n(x)). Denote by S_n (f,x) its partial sums and by E_n~q(f,x) its Euler (E, q)-means, i. e. E_n~q(f,x)=1/(1+q)~π sum from m=0 to n((?)q~(n-m)S_m(f,x)), with q≥0 (E_n~0≡S_n). In [1] Holland and Sahney proved the following theorem. THEOREM A Ifω(f,t) is the modulus of continuity of f∈C_(2π), then the degree of approximation of f by the (E,q)-means of f is givens by##特殊公式未编改  相似文献   

8.
Let{Xn;n≥1}be a sequence of i.i.d, random variables with finite variance,Q(n)be the related R/S statistics. It is proved that lim ε↓0 ε^2 ∑n=1 ^8 n log n/1 P{Q(n)≥ε√2n log log n}=2/1 EY^2,where Y=sup0≤t≤1B(t)-inf0≤t≤sB(t),and B(t) is a Brownian bridge.  相似文献   

9.
The Hausdorff dimensions of some refined irregular sets associated with β-expansions are determined for any β 1. More precisely, Hausdorff dimensions of the sets {x ∈ [0, 1) :lim inf(n→∞) S_n(x, β)/n= α_1, lim sup (n→∞) S_n(x, β)/n= α_2}, α_1, α_2≥0 are obtained completely, where S_n(x, β) =sum ε_k(x, β) from k=1 to n denotes the sum of the first n digits of the β-expansion of x. As an application, we present another concise proof of that the set of points x ∈ [0, 1) satisfying lim_(n→∞) S_n(x,β)/n does not exist is of full Hausdorff dimension.  相似文献   

10.
Let X_1, X_2,... be a sequence of independent random variables and S_n=sum X_1 from i=1 to n and V_n~2=sum X_1~2 from i=1 to n . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n/V_n converges to a standard normal distribution if and only if max1≤i≤n|X_i|/V_n → 0 in probability and the mean of X_1 is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max1≤i≤n|X_i|/V_n → 0 in probability, then these sufficient conditions are necessary.  相似文献   

11.
We determine the maximal value ofr with the following property. If the convex hull of a set inR 2 contains a unit circleB, then a subset of at most four points can be selected so that the convex hull of this subset contains the circle of radiusr concentric withB. That the result is sharp is shown by the example when the original set is the set of vertices of a regular pentagon circumscribed aroundB. Imre Bárány was partially supported by Hungarian National Science Foundation Grant Nos. 1907 and 1909. Aladár Heppes was partially supported by Hungarian National Science Foundation Grant No. 2583.  相似文献   

12.
Let be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H,‖⋅‖) with covariance operator Σ, and set Sn=X1+?+Xn, n?1. Let . We prove that, for any 1<r<3/2 and a>−d/2,
  相似文献   

13.
This paper deals with the continuity of the sharp constant K(T,X) with respect to the set T in the Jackson-Stechkin inequality $E(f,L) \leqslant K(T,X)\omega (f,T,X),$ , where E(f,L) is the best approximation of the function f ∈ X by elements of the subspace L ? X, and ω is a modulus of continuity, in the case where the space L 2( $\mathbb{T}^d $ , ?) is taken for X and the subspace of functions g ∈ L 2( $\mathbb{T}^d $ , ?), for L. In particular, it is proved that the sharp constant in the Jackson-Stechkin inequality is continuous in the case where L is the space of trigonometric polynomials of nth order and the modulus of continuity ω is the classical modulus of continuity of rth order.  相似文献   

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We study the gap structure of the partial order of projections of the Calkin algebra of a complex, separable, infinite-dimensional Hilbert space. We prove the existence of an analytic Hausdorff gap in this partial order. As a consequence we obtain that under Todorcevic’s Axiom and MA the gap spectrum of P(C(H)) is strictly bigger than the gap spectrum of P(ω)/Fin.  相似文献   

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The article considers the problem of dynamic decision under uncertainty in the state and in its dynamics. Game models and identification and decision methods are developed that ensure consistency and asymptotic stationarity of dynamic equilibria.This study was partially financed by the Russian Foundation of Basic Research (95-01-01137).Translated from Nelineinye Dinamicheskie Sistemy: Kachestvennyi Analiz i Upravlenie — Sbornik Trudov. No. 3. pp. 10–22, 1993.  相似文献   

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For 1/2<<1 fixed, letE (T) denote the error term in the asymptotic formula for . We obtain some new bounds forE (T), and an _-result which is the analogue of the strongest _-result in the classical Dirichlet divisor problem.  相似文献   

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