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1.
In this paper we find a second class of sequences of random numbers (x
n
)
n=1
∞
(the orbit of the ergodic adding machine) such that the corresponding sequences of zeros and ones 1[0,y](x
n) (n=1,2,...,N) satisfy Central Limit Theorems with extremely small standard deviationσ
N=O(√logN), instead ofO(√N), asN → ∞.
Dedicated to Professor Benjamin Weiss on the occasion of his 60th birthday. 相似文献
2.
René L. Schilling 《Probability Theory and Related Fields》1998,112(4):565-611
Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C
c
∞(ℝ
n
)⊂D(A) and A|C
c
∞(ℝ
n
) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|∞≤c(1+|ξ|2) and |Imp(x,ξ)|≤c
0Rep(x,ξ). We show that the associated Feller process {X
t
}
t
≥0 on ℝ
n
is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour
of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β∞
x
:={λ>0:lim
|ξ|→∞
|
x
−
y
|≤2/|ξ||p(y,ξ)|/|ξ|λ=0} or δ∞
x
:={λ>0:liminf
|ξ|→∞
|
x
−
y
|≤2/|ξ|
|ε|≤1|p(y,|ξ|ε)|/|ξ|λ=0}, and obtain a.s. (ℙ
x
) that lim
t
→0
t
−1/λ
s
≤
t
|X
s
−x|=0 or ∞ according to λ>β∞
x
or λ<δ∞
x
. Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27].
Received: 21 July 1997 / Revised version: 26 January 1998 相似文献
3.
We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n
u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2
n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31). 相似文献
4.
F. M. Mukhamedov 《Mathematical Notes》2000,67(1):81-86
In this paper an analog of the Blum-Hanson theorem for quantum quadratic processes on the von Neumann algebra is proved, i.e.,
it is established that the following conditions are equivalent:
Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 102–109, January, 2000. 相似文献
i) | P( t )x is weakly convergent tox 0; |
ii) | for any sequence {a n} of nonnegative integrable functions on [1, ∞) such that ∝ 1 ∞ a n(t)dt=1 for anyn and lim n→∞ ∥a n∥∞=0, the integral ∝ 1 ∞ a n(t)P( t )x dt is strongly convergent tox 0 inL 2(M, ϕ), wherex ɛM,P( t ) is a quantum quadratic process,M is a von Neumann algebra, andϕ is an exact normal state onM. |
5.
B. M. Bennett 《Annals of the Institute of Statistical Mathematics》1972,24(1):469-472
Summary The Wilcoxon statisticW
+ [3], [4] is well known as a test of the hypothesisH
0 of equality of bivariate means based onn pairs of observations (x, y). In this paper the distribution and limiting distribution ofW
+ are derived under the assumption of ‘logit’ alternativesH
0 of a trend in the probabilities associated with then successive paired differencesz
i
=x
i
−y
i
(1=1,...,n). 相似文献
6.
Artūras Dubickas 《Monatshefte für Mathematik》2009,137(2):271-284
We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2
n
+ ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence
contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( − 2)
n
+ ν
n
], n = 0, 1, 2, . . . , where ν
0, ν
1, ν
2, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x
1, x
2, x
3, . . . satisfies the recurrence relation x
n+d
= cx
n
+ F(x
n+1, . . . , x
n+d-1) for each n ≥ 1, where c ≠ 0 is an integer,
F(z1,...,zd-1) ? \mathbb Z[z1,...,zd-1],{F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],} and lim
n→ ∞|x
n
| = ∞, then the number |x
n
| is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation
modulo 3. 相似文献
7.
It is shown that if {y
n} is a block of type I of a symmetric basis {x
n} in a Banach spaceX, then {y
n} is equivalent to {x
n} if and only if the closed linear span [y
n] of {y
n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x
n,f
n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f
n] has a complemented subspace isomorphic tol
p (respectively,l
q, 1/p+1/q=1 when 1<p<+∞ andc
0 whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f
n] are obtained. We also obtain necessary and sufficient conditions such that [f
n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x
n} such that every symmetric block basic sequence of {x
n} spans a complemented subspace inX butX is not isomorphic to eitherc
0 orl
p, 1≤p<+∞. 相似文献
8.
V. M. Korchevsky 《Vestnik St. Petersburg University: Mathematics》2010,43(4):217-219
We investigate relationship between Kolmogorov–s condition and Petrov–s condition in theorems on the strong law of large numbers
for a sequence of independent random variables X
1, X
2, … with finite variances. The convergence (S
n
− ES
n
)/n → 0 holds a.s. (here, S
n
= Σ
k=1
n
X
k
), provided that Σ
n=1∞
DX
n
/n
2 < ∞ (Kolmogorov’s condition) or DS
n
= O(n
2/ψ(n)) for some positive non-decreasing function ψ(n) such that Σ1/(nψ(n)) < ∞ (Petrov’s condition). Kolmogorov’s condition is shown to follow from Petrov’s condition. Besides, under some additional
restrictions, Petrov’s condition, in turn, follows from Kolmogorov’s condition. 相似文献
9.
A sequence (μ
n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ
n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence.
Let ‖μ‖ denote the total mass ofμ andδ
0 denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in
the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible
probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague
convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e
−1] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x
1 + ... +x
n)/a
n, an → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ
0. If furthermore the ratiosa
n+1/a
n are bounded above and below by positive numbers, thenL(x)=P[|X
1|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea
n → ∞ so that the limit measure=βδ
0.
To the memory of Shlomo Horowitz
This research was partially supported by the National Science Foundation. 相似文献
10.
Alfred Lehman 《Israel Journal of Mathematics》1963,1(1):22-28
Circular symmetry is defined for ordered sets ofn real numbers: (y)=(y
1,...,y
n). Letf(x) be non-decreasing and convex forx≧0 and let (y) be given except in arrangement. The Σ
i
=1n
f(|y
i−y
i+1|) (wherey
n+1=y
1) is minimal if (and under some additional assumptions only if) (y) is arranged in circular symmetrical order.
Sponsored by the Mathematics Research Center, United States Army under Contract No. DA-11-022-ORD-2059, University of Wisconsin,
Madison. 相似文献
11.
We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P
ω(Y
0, λ
0)-solutions (where Y
0 is zero or ±∞ and −∞ ≤ λ0 ≤ +∞) of nonlinear differential equations y
(n) = α
0
p(t)φ(y), where α
0 ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y
0. 相似文献
12.
Frank Blume 《Israel Journal of Mathematics》1998,108(1):1-12
If (X,T) is a completely ergodic system, then there exists a positive monotone increasing sequence {a
n
}
n
1/∞
with lim
n
→∞a
n
=∞ and a positive concave functiong defined on [1, ∞) for whichg(x)/x
2 isnot integrable such that
for all nontrivial partitions α ofX into two sets. 相似文献
13.
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
14.
We consider an Abel equation (*)y’=p(x)y
2 +q(x)y
3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane)
is thaty
0=y(0)≡y(1) for any solutiony(x) of (*).
Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y
2 +εq(x)y
3
p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1)..
We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm
k
(1), wherem
k
(x)=∫
0
x
pk
(t)q(t)(dt),P(x)=∫
0
x
p(t)dt. We investigate the structure of zeroes ofm
k
(x) and generalize a “canonical representation” ofm
k
(x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric
center problem.
The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the
Minerva Foundation. 相似文献
15.
M. M. Sheremeta 《Ukrainian Mathematical Journal》1999,51(8):1296-1302
We establish the relation between the increase of the quantityM(σ,F) = ∣a
0∣ + ∑
n=1
∞
∣a
n
∣ exp (σλ
n
) and the behavior of sequences (|a
n
|) and (λ
n
), where (λ
n
) is a sequence of nonnegative numbers increasing to + ∞, andF(s) =a
0 + ∑
n=1
∞
a
n
e
sλn
,s=σ+it, is the Dirichlet entire series.
Lviv University, Lviv. Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 51, No. 8, pp. 1149–1153, August, 1999. 相似文献
16.
Roger D. Nussbaum 《Israel Journal of Mathematics》1991,76(3):345-380
Suppose thatE is a finite-dimensional Banach space with a polyhedral norm ‖·‖, i.e., a norm such that the unit ball inE is a polyhedron. ℝ
n
with the sup norm or ℝ
n
with thel
1-norm are important examples. IfD is a bounded set inE andT:D→D is a map such that ‖T(y)−T(z)‖≤ ‖y−z‖ for ally andz inE, thenT is called nonexpansive with respect to ‖·‖, and it is known that for eachx ∈D there is an integerp=p(x) such that lim
j→∞
T
jp
(x) exists. Furthermore, there exists an integerN, depending only on the dimension ofE and the polyhedral norm onE, such thatp(x)≤N: see [1,12,18,19] and the references to the literature there. In [15], Scheutzow has raised a question about the optimal
choice ofN whenE=ℝ
n
,D=K
n
, the set of nonnegative vectors in ℝ
n
, and the norm is thel
1-norm. We provide here a reasonably sharp answer to Scheutzow’s question, and in fact we provide a systematic way to generate
examples and use this approach to prove that our estimates are optimal forn≤24. See Theorem 2.1, Table 2.1 and the examples in Section 3. As we show in Corollary 2.3, these results also provide information
about the caseD=ℝ
n
, i.e.,T:ℝ
n
→ℝ
n
isl
1-nonexpansive. In addition, it is conjectured in [12] thatN=2
n
whenE=ℝ
n
and the norm is the sup norm, and such a result is optimal, if true. Our theorems here show that a sharper result is true
for an important subclass of nonexpansive mapsT:(ℝ
n
,‖ · ‖∞)→(ℝ
n
,‖ · ‖∞).
Partially supported by NSF DMS89-03018. 相似文献
17.
We investigate when the set of finite products of distinct terms of a sequence 〈x
n
〉
n=1∞ in a semigroup (S,⋅) is large in any of several standard notions of largeness. These include piecewise syndetic, central, syndetic, central*, and IP*. In the case of a “nice” sequence in (S,⋅)=(ℕ,+) one has that FS(〈x
n
〉
n=1∞) has any or all of the first three properties if and only if {x
n+1−∑
t=1
n
x
t
:n∈ℕ} is bounded from above.
N. Hindman acknowledges support received from the National Science Foundation via Grant DMS-0554803. 相似文献
18.
R. Choukri A. El Kinani A. Oukhouya 《Rendiconti del Circolo Matematico di Palermo》2007,56(2):235-243
We characterize locally convex topological algebrasA satisfying: a sequence (x
n) inA converges to 0 if, and only if, (x
n
2) converges to 0. We also show that a real Banach algebra such thatx
n
2+y
n
2→0 if, and only if,x
n → 0 andy
n → 0, for every sequences (x
n) and (y
n) inA, is isomorphic to, whereX is a compact space.
相似文献
19.
Let ƒ be a birational map of C
d
,and consider the degree complexity or asymptotic degree growth rate δ(ƒ) = limn → ∞ (deg(ƒn))1/n.We introduce a family of elementary maps, which have the form ƒ = L o J, where L is (invertible) linear, and J(x
1
−1
,..., xd) = (x
1
−1
,...,x
d
−1
.We develop a method of regularization and show how it can be used to compute δ for an elementary map. 相似文献
20.
K.H. Kwon D.W. Lee F. Marcellán S.B. Park 《Annali di Matematica Pura ed Applicata》2001,180(2):127-146
Given an orthogonal polynomial system {Q
n
(x)}
n=0
∞, define another polynomial system by where α
n
are complex numbers and t is a positive integer. We find conditions for {P
n
(x)}
n=0
∞ to be an orthogonal polynomial system. When t=1 and α1≠0, it turns out that {Q
n
(x)}
n=0
∞ must be kernel polynomials for {P
n
(x)}
n=0
∞ for which we study, in detail, the location of zeros and semi-classical character.
Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001 相似文献