共查询到20条相似文献,搜索用时 46 毫秒
1.
Ryoichi Shimizu 《Annals of the Institute of Statistical Mathematics》1986,38(1):195-204
Summary LetX be a positive random variable with the survival function
and the densityf. LetX have the moments μ=E(X) and μ2=E(X
2) and put ε=|1-μ2/2μ2|. Put
and
. It is proved that the following inequalities hold:
, for allx>0, ifq(x) is monotone and that
, ifq
1
(x) is monotone. It is also shown that Brown's inequality
which holds wheneverq
1
(x) is increasing is not valid in general whenq
1 is decreasing.
The Institute of Statistical Mathematics 相似文献
2.
Vidmantas Bentkus 《Israel Journal of Mathematics》2007,158(1):1-17
Let M
n
= X
1 + ⋯ + X
n
be a martingale with bounded differences X
m
= M
m
− M
m
−1 such that ℙ{a
m
− σ
m
≤ X
m
≤ a
m
+ σ
m
} = 1 with nonrandom nonnegative σ
m
and σ(X
1, …, X
m
−1)-measurable random variables a
m
. Write σ
2 = σ
1
2
+ ⋯ + σ
n
2
. Let I(x) = 1 − Φ(x), where Φ is the standard normal distribution function. We prove the inequalities
with a constant c such that 3.74 … ≤ c ≤ 7.83 …. The result yields sharp bounds in some models related to the measure concentration. In the case where all a
m
= 0 (or a
m
≤ 0), the bounds for constants improve to 3.17 … ≤ c ≤ 4.003 …. The inequalities are new even for independent X
1, …, X
n
, as well as for linear combinations of independent Rademacher random variables.
Research supported by Max Planck Institute for Mathematics, Bonn 相似文献
3.
We consider a positive distribution Φ such that Φ defines a probability measure μ=μ
Φ on the dual of some real nuclear Fréchet space. A large deviation principle is proved for the family {μ
n
,n≥1} where μ
n
denotes the image measure of the product measure μ
Φ
n
under the empirical distribution function L
n
. Here the rate function I is defined on the space ℱ′
θ
(N′)+ and agrees with the relative entropy function
. As an application, we cite the Gibbs conditioning principle which describes the limiting behaviour as n tends to infinity of the law of k tagged particles Y
1,…,Y
k
under the constraint that L
n
Y
belongs to some subset A
0.
相似文献
4.
S. John 《Annals of the Institute of Statistical Mathematics》1973,25(1):363-372
Summary Let the two alternative populationsP
1 andP
2 from which the individual with measurements χ may have come beN(μ(1), Σ) andN(μ(2), Σ). Then the classification rule with minimum risk is to assign the individual toP
1 orP
2 according as (μ(2)-μ(1))′Σ-1
x≶(1/2)(μ(2)-μ(1))′Σ-1(μ(1)+μ(2))+c wherec is a constant depending on the prior probabilities ofP
1 andP
2 and the costs of the two kinds of misclassification. The probability of misclassifying an individual fromP
2 by this rule is π21=Φ(-δ/2+cδ-1), where Φ(.) is the distribution function of anN(0, 1) and
. (Since we are free to choose which population we shall callP
2, it is not necessary to consider separately the probability of misclassifying an individual fromP
1.) LetP
21 denote the probability of misclassification of an individual fromP
2 by the rule derived from the one mentioned by fixing μ(1), μ(2) and Σ at estimates
andV and letP
21
*
be the probability of misclassification of an individual fromP
2 when the classification rule is the one with minimum risk among those based on
. The fiducial distributions of π21,P
21 andP
21
*
are determined. Point estimates and confidence intervals for π21,P
21 andP
21
*
are derived. Only easily available tables are needed to make fiducial inferences. An incidental result of some interest elsewhere
as well is the distribution of a linear combination of a chi and an independent normal variable. 相似文献
5.
We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions Φ1, Φ2 and Φ, generating corresponding Orlicz spaces, satisfy the estimate ${\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)}We show that the result on multipliers of Orlicz spaces holds in general. Namely, under the assumption that three Young functions
Φ1, Φ2 and Φ, generating corresponding Orlicz spaces, satisfy the estimate F-1(u) £ C F1-1(u) F2-1(u){\Phi^{-1}(u) \leq C \Phi_1^{-1}(u)\, \Phi_2^{-1}(u)} for all u > 0, we prove that if the pointwise product xy belongs to L
Φ(μ) for all y ? LF1(m){y \in L^{\Phi_1}(\mu)}, then x ? LF2(m){x \in L^{\Phi_2}(\mu)}. The result with some restrictions either on Young functions or on the measure μ was proved by Maligranda and Persson (Indag. Math. 51 (1989), 323–338). Our result holds for any collection of three Young functions satisfying the above estimate and for an arbitrary
complete σ-finite measure μ. 相似文献
6.
Yehoram Gordon 《Israel Journal of Mathematics》1969,7(2):151-163
Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ
p(X) as inf{Σ
i
=1/m
|x*(x
i)|p
p Σ
i
=1/m
‖x
i‖p
p]1
p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x
1,x
2, …,x
m} ⊂X such that Σ
i
=1/m
‖x
i‖>0. It follows immediately from [2] thatμ
p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ
p(X) for various spaces, and obtain some asymptotic estimates ofμ
p(X) for general finite dimensional Banach spaces.
This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof.
A. Dvoretzky and Prof. J. Lindenstrauss. 相似文献
7.
Let Bσ,p,1 <-p<-∞, be the set of all functions from L8(R) which can be continued to entire functions of exponential type <-σ. The well known Shannon sampling theorem and its generalization [1] state that every f∈Bσ,p, 1<p<∞, can be represented as
$f(x) = \mathop \Sigma \limits_{j \in z} f(j\pi /\sigma )\tfrac{{sin\sigma (x - j\pi /\sigma )}}{{\sigma (x - j\pi /\sigma )}}, \sigma > 0$f(x) = \mathop \Sigma \limits_{j \in z} f(j\pi /\sigma )\tfrac{{sin\sigma (x - j\pi /\sigma )}}{{\sigma (x - j\pi /\sigma )}}, \sigma > 0 相似文献
8.
We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier
series: For all measure-preserving flows (X,μ,T
t
) and f∈L
p
(X,μ), there is a set X
f
⊂X of probability one, so that for all x∈X
f
,
9.
S. J. Montgomery-Smith 《Israel Journal of Mathematics》1989,67(1):123-128
We show that the canonical embeddingC(K) →L
Φ(μ) has Gaussian cotypep, where μ is a Radon probability measure onK, and Φ is an Orlicz function equivalent tot
p(logt)
p/2 for larget. 相似文献
10.
G. Schlüchtermann 《manuscripta mathematica》1991,73(1):397-409
A sufficient condition is given when a subspaceL⊂L
1(μ,X) of the space of Bochner integrable function, defined on a finite and positive measure space (S, Φ, μ) with values in a Banach spaceX, is locally uniformly convex renormable in terms of the integrable evaluations {∫
A
fdμ;f∈L}. This shows the lifting property thatL
1(μ,X) is renormable if and only ifX is, and indicates a large class of renormable subspaces even ifX does not admit and equivalent locally uniformly convex norm. 相似文献
11.
Coenraad C.A. Labuschagne Valeria Marraffa 《Central European Journal of Mathematics》2010,8(1):148-157
Spaces of cone absolutely summing maps are generalizations of Bochner spaces L
p
(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $
\mathcal{L}^1 \left[ {\sum ,cbf(X)} \right]
$
\mathcal{L}^1 \left[ {\sum ,cbf(X)} \right]
of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L
1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $
\mathcal{L}^1 \left[ {\sum ,cbf(X)} \right]
$
\mathcal{L}^1 \left[ {\sum ,cbf(X)} \right]
, and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map.
We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably
bounded set-valued functions. 相似文献
12.
Nicholas A. Loehr Gregory S. Warrington Herbert S. Wilf 《Israel Journal of Mathematics》2004,143(1):141-156
We study the polynomial
, where ω is a primitivepth root of unity. This polynomial arises in CR geometry [1]. We show that it is the determinant of thep×p circulant matrix whose first row is (1, −x,0,…,0,−y,0,…,0), the −y being in positionq+1. Therefore, the coefficients of this polynomial Φ are integers that count certain classes of permutations. We show that
all of the permutations that contribute to a fixed monomialx
rys in Φ have the same sign, and we determine that sign. We prove that a monomialx
rys appears in Φ if and only ifp dividesr+sq. Finally, we show that the size of the largest coefficient of the monomials in Φ grows exponentially withp, by proving that the permanent of the circulant whose first row is (1, 1, 0, …, 0, 1, 0, …, 0) is the sum of the absolute
values of the monomials in the polynomial Φ.
Supported by NSF Postdoctoral research grants. 相似文献
13.
M. M. Rao 《Israel Journal of Mathematics》1968,6(2):133-149
In this paper the clasical Hausdorff-Young theorem, which states that iff ∈L
p, 1≦p≦2, on the line and
is its Fourier transform, then
whereq
−1+p
−1=1, is extended in two ways for certain Orlicz spacesL
Φ. IfL
Φ is based on (G, μ), (1) an arbitrary compact topological group with Haar measure, and (2) a locally compact abelian topological group andμ is again the Haar measure, then the above inequality is extended to these cases. Various other related results and remarks
are also included.
Dedicated to the memory of my nephew, K. Ramakrishna, who appeared to be so brilliant.
This research was supported by the NSF Grants GP-5921 and GP-7678. 相似文献
14.
A. I. Pavlov 《Mathematical Notes》2000,68(3):370-377
The main result of this paper is the following theorem. Suppose thatτ(n) = ∑
d|n
l and the arithmetical functionF satisfies the following conditions:
15.
Michael Lin 《Israel Journal of Mathematics》1970,8(4):357-366
LetP be a conservative and ergodic Markov operator onL
1(X, Σ,m). We give a sufficient condition for the existence of a decompositionA
f
↑X such that for 0≦f, g ∈L
∞ (A
j
) and any two probability measuresμ andν weaker thanm
, whereλ is theσ-finite invariant measure (which necessarily exists). Processes recurrent in the sense of Harris are shown to have this decomposition,
and an analytic proof of the convergence of
is deduced for such processes.
This paper is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the direction of Professor
S. R. Foguel, to whom the author is grateful for his helpful advice and kind encouragement. 相似文献
16.
Robert Chen 《Israel Journal of Mathematics》1976,24(3-4):244-259
LetX be a non-empty set,H= X{su\t8, \gs = \lj{in1}x\lj{in2}x,σ=γ
1×γ
2×… be an independent strategy onH, and {Y
n} be a sequence of coordinate mappings onH. The following strong law in a finitely additive setting is proved: For some constantr≧1, if \GS
n=1
\t8
{\GS(\vbY
n
\vb2r
)n
1+n
< \t8 andσ(Y
n)=0 for alln=1, 2, …, then \1n\gS{inj-1}/{sun} Y{inj}Y
jconverges to 0 withσ-measure 1 asn → ∞. The theorem is a generalization of Chung’s strong law in a coordinate representation process. Finally, Kolmogorov’s
strong law in a finitely additive setting is proved by an application of the theorem.
This research was based in part on the author’s doctoral dissertation submitted to the University of Minnesota, and was written
with the partial support of the United States Army Grant DA-ARO-D-31-124-70-G-102. 相似文献
17.
Jie-Cheng Chen 《Israel Journal of Mathematics》1993,81(1-2):193-202
In this paper, we get a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL
Φ(R
n, v(x)dx) to weak-L
Φ(R
+
n+1
,dμ), where Φ is anN-function satisfying the Δ2-condition. We also find a necessary and sufficient condition on the weights (μ,v) for the Poisson integral operator to be bounded fromL
Φ(R
n,v(x)dx) toL
Φ(R
+
n+1
,dμ) under some additional condition.
Partially supported by NNSF of P.R. China 相似文献
18.
Y. C. Wang 《Acta Mathematica Hungarica》2012,135(3):248-269
Let Hk\mathcal{H}_{k} denote the set {n∣2|n,
n\not o 1 (mod p)n\not\equiv 1\ (\mathrm{mod}\ p) ∀ p>2 with p−1|k}. We prove that when
X\frac1120(1-\frac12k) +e\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{2k}\right) +\varepsilon}\leqq H\leqq X, almost all integers
n ? \allowbreak Hk ?(X, X+H]n\in\allowbreak {\mathcal{H}_{k} \cap (X, X+H]} can be represented as the sum of a prime and a k-th power of prime for k≧3. Moreover, when
X\frac1120(1-\frac1k) +e\leqq H\leqq XX^{\frac{11}{20}\left(1-\frac{1}{k}\right) +\varepsilon}\leqq H\leqq X, almost all integers n∈(X,X+H] can be represented as the sum of a prime and a k-th power of integer for k≧3. 相似文献
19.
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
20.
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 |