共查询到20条相似文献,搜索用时 15 毫秒
1.
Ignacy Kotlarski 《Annali di Matematica Pura ed Applicata》1966,74(1):129-134
Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x1, x2, …, xn be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals fk(t) = E[ei(t,x
k)], (k=1, 2, …, n): let y1=x1 + xn, y2=x2 + xn, …, yn−1=xn−1 + xn.
If the characteristic functional f(t1, t2, …, tn−1) of the random variables (y1, y2, …, yn−1) does not vanish, then the joint distribution of (y1, y2, …, yn−1) determines all the distributions of x1, x2, …, xn up to change of location. 相似文献
2.
Extremal probabilities for Gaussian quadratic forms 总被引:1,自引:0,他引:1
Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf
Q
P(Q≤x)=P(χ2
n
/n≤x), where χ
n
2
is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z
1
,Z
2
are independent standard Gaussian rv's, then sup0≤λ≤1/2
P{λZ
1
2
+(1−λ)Z
2
2
≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic
quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes.
Received: 24 June 2002 / Revised version: 26 January 2003
Published online: 15 April 2003
Research supported by NSA Grant MDA904-02-1-0091
Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10 相似文献
3.
J. C. Gupta 《Proceedings Mathematical Sciences》2000,110(4):415-430
Let G
n,k
be the set of all partial completely monotone multisequences of ordern and degreek, i.e., multisequencesc
n(β1,β2,…, β
k
), β1,β2,…, βk
= 0,1,2,…, β1+β2 + … +β
k
≤n,c
n(0,0,…, 0) = 1 and
whenever β0 ≤n - (β1 + β2 + … + β
k
) where Δc
n(β1,β2,…, β
k
) =c
n(β1 + 1, β2,…, β
k
)+c
n(β1,β2+1,…, β
k
)+…+c
n (β1,β2,…, β
k
+1) -c
n(β1,β2,…, 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}(β1,β2,…, 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
0(β1, 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
4.
Sofiya Ostrovska 《Proceedings Mathematical Sciences》2007,117(4):485-493
Let φ be a power series with positive Taylor coefficients {a
k
}
k=0∞ and non-zero radius of convergence r ≤ ∞. Let ξ
x
, 0 ≤ x < r be a random variable whose values α
k
, k = 0, 1, …, are independent of x and taken with probabilities a
k
x
k
/φ(x), k = 0, 1, ….
The positive linear operator (A
φ
f)(x):= E[f(ξ
x
)] is studied. It is proved that if E(ξ
x
) = x, E(ξ
x
2) = qx
2 + bx + c, q, b, c ∈ R, q > 0, then A
φ
reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1. 相似文献
5.
E. G. Goluzina 《Journal of Mathematical Sciences》2008,150(3):2005-2012
The paper studies the regions of values of the systems {f(z1), f(r1), f(r2),…, f(rn)} and {f(r1), f(r2),…, f (rn)}, where n 2; z1 is an arbitrary fixed point of the disk U = {z: |z| < 1} with Im z1 ≠ 0; rj are fixed numbers, 0 < rj < 1, j = 1, 2,…, n; f ∈ T, and the class T consists of the functions f(z), f(0) = 0, f′(0) = 1, regular in the disk U and
satisfying the condition Im f(z) · Imz > 0 for Im z ≠ 0. As an implication, the region of values of f(z1) in the subclass of functions f ∈ T with prescribed values f(rj) (j = 1, 2,…, n) is determined. Bibliography: 12 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 5–16. 相似文献
6.
A. Dubickas 《Journal of Mathematical Sciences》2006,137(2):4654-4657
Let g and m be two positive integers, and let F be a polynomial with integer coefficients. We show that the recurrent sequence
x0 = g, xn = x
n−1
n
+ F(n), n = 1, 2, 3,…, is periodic modulo m. Then a special case, with F(z) = 1 and with m = p > 2 being a prime number,
is considered. We show, for instance, that the sequence x0 = 2, xn = x
n−1
n
+ 1, n = 1, 2, 3, …, has infinitely many elements divisible by every prime number p which is less than or equal to 211 except
for three prime numbers p = 23, 47, 167 that do not divide xn. These recurrent sequences are related to the construction of transcendental numbers ζ for which the sequences [ζn!], n = 1, 2, 3, …, have some nice divisibility properties. Bibliography: 18 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 76–82. 相似文献
7.
Ermanno Marchionna 《Annali di Matematica Pura ed Applicata》1954,37(1):265-290
Sunto Si dimostra che: data in Sr-1 un'ipersuperficie generale F(x1, x2, …, xr-1)=0 e su di essa una varietà D che sia di diramazione per una funzione u(x1, x2, …, xr-1) — definita da un'ipersuperficie generale Aμ(x1, x2, …, xr-1, u) =0 d'ordine μ >4 (di Sr) — qualunque funzione u' dei punti di F diramata dalla D è birazionalmente identica alla u. La dimostrazione viene ricondotta
al caso delle curve multiple, in virtù di un criterio d'identità qui appositamente stabilito, ed è basata sulla costruzione
della Riemanniana di una curva multipla generale. 相似文献
8.
Let Γ r,n-r denote the infimum of all numbers Γ>0 such that for any real indefinite quadraticQ inn variables of type (r, n?r), determinantD≠0 and real numbersc 1,…,c n there exist (x 1,…,x n )≡(c 1,…,c n ) (mod 1) satisfying $$0< Q(x_1 ,...,x_n ) \leqslant (\Gamma \left| D \right|)^{1/n} .$$ . All the values of Γ r,3 are known except Γ1,4. It is shown that $$8 \leqslant \Gamma _{1,4} \leqslant 16.$$ . 相似文献
9.
A. V. Zheleznyak 《Vestnik St. Petersburg University: Mathematics》2009,42(4):269-274
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
10.
D. M. Smirnov 《Algebra and Logic》1998,37(2):92-100
The dimension of a variety V of algebras is the greatest length of a basis (i.e., of an independent generating set) for an
SC-theory SC(V), consisting of strong Mal'tsev conditions satisfied in V. The variety V is assumed infinite-dimensional if
the lengths of the bases in SC(V) are not bounded. A simple algorithm is found for constructing a variety of any finite dimension
r≥1. Using the sieve of Eratosthenes, r distinct primes p1, p2,…, pr are written and their product n=p1p2…pr computed. The variety Gn of algebras (A, f) with one n-ary operation satisfying the identity f(x1, x2,…,xn)=f(x2,…,xn, x1) has, then, dimension r.
Translated fromAlgebra i Logika, Vol. 37, No. 2, pp. 167–180, March–April, 1998. 相似文献
11.
Mohamed S. El-Mosalamy 《Israel Journal of Mathematics》1986,56(3):345-348
In this paper we give a simple proof of Collin’s theorem concerning free subgroups ofC(4),T(4) groups. Our proof actually shows that a slenderT(4) presentation 〈x
1,x
2, …,x
n
;r〉 has a free subgroup of rank 2 provided there is a subset {a, b, c} of {x
1,x
2, …,x
n
} with the property that any non-empty freely reduced word ina, b, c equal to 1 inG has a subword of length 2 contained in an element ofr*. 相似文献
12.
This paper considers thefinitary reconstruction of an ergodic measure preserving transformationT of a complete separable metric spaceX from a single trajectoryx, Tx, …, or more generally, from a suitable reconstruction sequence x=x
1,x
2, … withx
i∈X. Ann-sample reconstruction is a functionT
n: X
n+1 →X; the map
(·;x
1, …,x
n)is treated as an estimate ofT(·) based on then initial elements of x. Given a reference probability measureμ
0 and constantM>1, functionsT
1,T
2, … are defined, and it is shown that for everyμ with 1/M≤dμ/dμ
0≤M, everyμ-preserving transformationT, and every reconstruction sequence x forT, the estimates
(·;x
1, …,x
nconverge toT in the weak topology.
For the family of interval exchange transformations of [0, 1] a simple family of estimates is described and shown to be consistent
both pointwise and in the strong topology. However, it is also shown that no finitary estimation scheme is consistent in the
strong topology for the family of all ergodic Lebesgue measure preserving transformations of the unit interval, even if x
is assumed to be a generic trajectory ofT.
Supported in part by NSF Grant DMS-9501926. 相似文献
13.
Felix Schlenk 《Israel Journal of Mathematics》2003,138(1):215-252
We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid.
It is first proved that under the conditionr
n
2
≤2r
1
2
the symplectic ellipsoidE(r
1,…,r
n)with radiir
1≤…≤r
ndoes not symplectically embed into a ball of radius strictly smaller thanr
n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected
symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids. 相似文献
14.
E. G. Goluzina 《Journal of Mathematical Sciences》1998,89(1):958-966
Let TR be the class of functions
that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z0), f(r)} for z0 ∈ ℝ, r∈(-1,1) is studied. The sets Dr={f(z0):f∈TR, f(r)=a} for −1≤r≤1 and Δr={(c2, c3): f ∈ TR, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)−2, r(1−r)−2) is an arbitrary fixed number. Bibliography: 11 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79. 相似文献
15.
E. Rips 《Israel Journal of Mathematics》1981,39(4):326-340
Letf
1, …,f
n be free generators of a free groupF. We consider the equation [z
1, …,z
n]ω. where ω and ω′ indicate the disposition of brackets in the higher commutators [z
1, …,z
n]ω and [f
1, …,f
n]ω. We give a necessary and sufficient condition on ω and ω′ for the existence of solutions of this equation. It is also shown
that for any solutionz
1=r1, …,z
z=r
n we have <r
1, …,r
n>=〈f
1, …f
n〉. 相似文献
16.
Daniel Wulbert 《Israel Journal of Mathematics》2001,126(1):363-380
LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L
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 n≤m, such that for all g ∈ G,
相似文献
17.
For x = (x
1, x
2, …, x
n
) ∈ (0, 1 ]
n
and r ∈ { 1, 2, … , n}, a symmetric function F
n
(x, r) is defined by the relation
Fn( x,r ) = Fn( x1,x2, ?, xn;r ) = ?1 \leqslant1 < i2 ?ir \leqslant n ?j = 1r \frac1 - xijxij , {F_n}\left( {x,r} \right) = {F_n}\left( {{x_1},{x_2}, \ldots, {x_n};r} \right) = \sum\limits_{1{ \leqslant_1} < {i_2} \ldots {i_r} \leqslant n} {\prod\limits_{j = 1}^r {\frac{{1 - {x_{{i_j}}}}}{{{x_{{i_j}}}}}} }, 相似文献
18.
For distinct points x1,x2,…,xn in ℛ (the reals), letϕ[x1, x2,…,xn] denote the divided difference ofϕ. In this paper, we determine the general solutionϕ,g: ℛ → ℛ of the functional equationϕ[x1,x2,…,xn] =g(x1,+ x2 + … + xn) for distinct x1,x2,…, xn in ℛ without any regularity assumptions on the unknown functions. 相似文献
19.
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
20.
The method of cyclic relaxation for the minimization of a function depending on several variables cyclically updates the value
of each of the variables to its optimum subject to the condition that the remaining variables are fixed.
We present a simple and transparent proof for the fact that cyclic relaxation converges linearly to an optimum solution when
applied to the minimization of functions of the form
for a
i,j
,b
i
,c
i
∈ℝ≥0 with max {min {b
1,b
2,…,b
n
},min {c
1,c
2,…,c
n
}}>0 over the n-dimensional interval [l
1,u
1]×[l
2,u
2]×⋅⋅⋅×[l
n
,u
n
] with 0<l
i
<u
i
for 1≤i≤n. Our result generalizes several convergence results that have been observed for algorithms applied to gate- and wire-sizing
problems that arise in chip design. 相似文献
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