共查询到20条相似文献,搜索用时 31 毫秒
1.
Hans-Peter Scheffler 《Journal of Mathematical Analysis and Applications》2003,288(1):285-298
Let X1,X2,… be i.i.d. random variables with distribution μ and with mean zero, whenever the mean exists. Set Sn=X1+?+Xn. In recent years precise asymptotics as ε↓0 have been proved for sums like ∑n=1∞n−1P{|Sn|?εn1/p}, assuming that μ belongs to the (normal) domain of attraction of a stable law. Our main results generalize these results to distributions μ belonging to the (normal) domain of semistable attraction of a semistable law. Furthermore, a limiting case new even in the stable situation is presented. 相似文献
2.
Shaofang Hong 《Linear algebra and its applications》2008,428(4):1001-1008
Let a,b and n be positive integers and the set S={x1,…,xn} of n distinct positive integers be a divisor chain (i.e. there exists a permutation σ on {1,…,n} such that xσ(1)|…|xσ(n)). In this paper, we show that if a|b, then the ath power GCD matrix (Sa) having the ath power (xi,xj)a of the greatest common divisor of xi and xj as its i,j-entry divides the bth power GCD matrix (Sb) in the ring Mn(Z) of n×n matrices over integers. We show also that if a?b and n?2, then the ath power GCD matrix (Sa) does not divide the bth power GCD matrix (Sb) in the ring Mn(Z). Similar results are also established for the power LCM matrices. 相似文献
3.
Fuqing Gao 《Journal of Mathematical Analysis and Applications》2003,285(1):1-7
Let X0,X1,… be i.i.d. random variables with E(X0)=0, E(X20)=1 and E(exp{tX0})<∞ for any |t|<t0. We prove that the weighted sums V(n)=∑j=0∞aj(n)Xj, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums. 相似文献
4.
Mohammad Sal Moslehian Dorian Popa 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(9):2792-2799
Suppose that X is a sequentially complete Hausdorff locally convex space over a scalar field K, V is a bounded subset of X, (an)n≥0 is a sequence in K?{0} with the property lim infn→∞|an|>1, and (bn)n≥0 is a sequence in X. We show that for every sequence (xn)n≥0 in X satisfying
5.
Li Xin Zhang 《数学学报(英文版)》2008,24(4):631-646
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
6.
Let (a,b)∈Z2, where b≠0 and (a,b)≠(±2,−1). We prove that then there exist two positive relatively prime composite integers x1, x2 such that the sequence given by xn+1=axn+bxn−1, n=2,3,… , consists of composite terms only, i.e., |xn| is a composite integer for each n∈N. In the proof of this result we use certain covering systems, divisibility sequences and, for some special pairs (a,±1), computer calculations. The paper is motivated by a result of Graham who proved this theorem in the special case of the Fibonacci-like sequence, where (a,b)=(1,1). 相似文献
7.
L. B. Klebanov 《Mathematical Notes》1973,13(6):531-532
Let (X1, ..., Xn) be a random vector with independent components. It is proven in this paper that, under certain restrictions, the distributions of the pairS
1=sup (a
1X1, ..., anXn) andS
2=sup (b1X1,...,bnXn) univocally define the distribution function of the components Xj.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 889–892, June, 1973. 相似文献
8.
Let X, X1, X2, … be i.i.d. random variables with nondegenerate common distribution function F, satisfying EX = 0, EX2 = 1. Let Xi and Mn = max{Xi, 1 ≤ i ≤ n }. Suppose there exists constants an > 0, bn ∈ R and a nondegenrate distribution G (y) such that Then, we have almost surely, where f (x, y) denotes the bounded Lipschitz 1 function and Φ(x) is the standard normal distribution function (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
This paper deals with the quality of approximative solutions for the Subset-Sum-Maximization-Problem maximize $$\sum\limits_{i = l}^n {a_i x_i } $$ subject to $$\sum\limits_{i = l}^n {a_i x_i } \leqslant b$$ wherea l,...,an,bεR+ andx l,...xnε{0,1}. produced by certain heuristics of a Greedy-type. Every heuristic under consideration realizes a feasible solution (x 1, ..., xn) whose objective value is less or equal the optimal value, which is of course not greater thanb. We use the gap between capacityb and realized value as an upper bound for the error made by the heuristic and as a criterion for quality. Under the stochastic model:a 1, ..., an, b independent,a 1...,an uniformly distributed on [0, 1], b uniformly distributed on [0,n] we derive the gap-distributions and the expected size of the gaps. The analyzed algorithms include four algorithms which can be done in linear time and four heuristics which require sorting, which means that they are done inO(nlnn) time. 相似文献
10.
Let X1, ... , Xn be i.i.d. integral valued random variables and Sn their sum. In the case when X1 has a moderately large tail of distribution, Deshouillers, Freiman and Yudin gave a uniform upper bound for max k ∊ ℤ Pr{Sn = k} (which can be expressed in term of the Lévy Doeblin concentration of Sn), under the extra condition that X1 is not essentially supported by an arithmetic progression. The first aim of the paper is to show that this extra condition cannot be simply ruled out. Secondly, it is shown that if X1 has a very large tail (larger than a Cauchy-type distribution), then the extra arithmetic condition is not sufficient to guarantee a uniform upper bound for the decay of the concentration of the sum Sn. Proofs are constructive and enhance the connection between additive number theory and probability theory.À Jean-Louis Nicolas, avec amitié et respect2000 Mathematics Subject Classification: Primary—60Fxx, 60Exx, 11Pxx, 11B25 相似文献
11.
Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables 总被引:1,自引:0,他引:1
Ye JIANG Li Xin ZHANG 《数学学报(英文版)》2006,22(3):781-792
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
12.
Uwe Einmahl 《Journal of Theoretical Probability》2007,20(4):901-915
Let X 1,X 2,… be a sequence of i.i.d. mean zero random variables and let S n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? n→∞|S n |/c n =α 0<∞ for a regular normalizing sequence {c n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting. 相似文献
13.
Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d W (·, ·) associated to the generating set {a, b, a ?1, b ?1}. Letting B n = {x ∈ ?: d W (x, e ?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ?1 b ?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B n → X is at least a constant multiple of (log n)1/q , an asymptotically optimal estimate as n → ∞. 相似文献
14.
《Journal of Computational and Applied Mathematics》1999,107(1):1-20
We propose a method of constructing orthogonal polynomials Pn(x) (Krall's polynomials) that are eigenfunctions of higher-order differential operators. Using this method we show that recurrence coefficients of Krall's polynomials Pn(x) are rational functions of n. Let Pn(a,b;M)(x) be polynomials obtained from the Jacobi polynomials Pn(a,b)(x) by the following procedure. We add an arbitrary concentrated mass M at the endpoint of the orthogonality interval with respect to the weight function of the ordinary Jacobi polynomials. We find necessary conditions for the parameters a,b in order for the polynomials Pn(a,b;M)(x) to obey a higher-order differential equation. The main result of the paper is the following. Let a be a positive integer and b⩾−1/2 an arbitrary real parameter. Then the polynomials Pn(a,b;M)(x) are Krall's polynomials satisfying a differential equation of order 2a+4. 相似文献
15.
Paul Sablonnière 《Numerical Algorithms》1992,3(1):401-409
LetL(α, r) denote the class of complex sequences (x n) having an asymptotic expansion of type $$x_n \sim \sum\limits_{i \ge 0} {c_i n^{ - (\alpha + ri)} } , c_0 \ne 0,\alpha > 0,r > 0.$$ We describe the asymptotic behaviour of sequences obtained by applying to (x n) some specific modified versions of the iterated Δ2 transform, the θ2 transform and some combinations of them. In this paper, we study the particular casesr=1 andr=1/2, which are the most useful in practice. The results are also valid for sequences (S n) whose error sequence (x n) defined byx n=S n?S, S=limS n, belongs to someL(α, r). 相似文献
16.
William B. Poucher 《Discrete Mathematics》1977,18(3):291-298
The pair (P, p) is a (partial) (n, b)-PBD if (P, p) isa (partial) pairwise balanced design with the property that |P| = n and each block in p has exactly b elements. The following theorems are proved.Theorem. If (P, p) is an (n, b)-PBD and n > b ? 4, then (P, p) has an isomorphic disjoint mate. (Theorem 2.3)Theorem. Suppose k and b are positive integers and b ? 5. There is a constant C(k, b) such that if (P, p) is an (n, b)-PBD and n > C(k, b), then there exist k mutually disjoint isomorphic mates of (P, p). (Theorem 2.2)Theorem. Suppose k and b are positive integers, k ? 2 and b ? 5. If (P, p1, (P, p2),…, (P, pk) is a collection of partial (|P|, b)-PBD's, there exist k (n, b)-PBD's (X, x1), (X, x2), …, (X, xk) such that (P, p1) is embedded in (X, x1) and for i ≠ j, p1 ∩ p1 = x1 ∩ x1. Additionally the existence of certain collections valuable in embedding is explored. (Theorem 4.10) 相似文献
17.
In this paper, we introduce a saddlepoint approximation method for higher-order moments like E(S − a)+
m
, a>0, where the random variable S in these expectations could be a single random variable as well as the average or sum of some i.i.d random variables, and
a > 0 is a constant. Numerical results are given to show the accuracy of this approximation method. 相似文献
18.
Michel Weber 《Indagationes Mathematicae》2007,18(2):281-293
Let ?= {?i,i ≥1} be a sequence of independent Bernoulli random variables (P{?i = 0} = P{?i = 1 } = 1/2) with basic probability space (Ω, A, P). Consider the sequence of partial sums Bn=?1+...+?n, n=1,2..... We obtain an asymptotic estimate for the probability P{P-(Bn) > >} for >≤ne/log log n, c a positive constant. 相似文献
19.
Divisibility of determinants of power GCD matrices and power LCM matrices on finitely many quasi-coprime divisor chains 总被引:1,自引:0,他引:1
Let a, n ? 1 be integers and S = {x1, … , xn} be a set of n distinct positive integers. The matrix having the ath power (xi, xj)a of the greatest common divisor of xi and xj as its i, j-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (Sa). Similarly we can define the ath power LCM matrix [Sa]. We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S1 ∪ ? ∪ Sk, where k ? 1 is an integer and all Si (1 ? i ? k) are divisor chains such that (max(Si), max(Sj)) = gcd(S) for 1 ? i ≠ j ? k. In this paper, we first obtain formulae of determinants of power GCD matrices (Sa) and power LCM matrices [Sa] on the set S consisting of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. Using these results, we then show that det(Sa)∣det(Sb), det[Sa]∣det[Sb] and det(Sa)∣det[Sb] if a∣b and S consists of finitely many quasi-coprime divisor chains with gcd(S) ∈ S. But such factorizations fail to be true if such divisor chains are not quasi-coprime. 相似文献
20.
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,