共查询到20条相似文献,搜索用时 93 毫秒
1.
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
2.
We consider the random variable ζ = ξ1ρ+ξ2ρ2+…, where ξ1, ξ2, … are independent identically distibuted random variables taking the values 0 and 1 with probabilities P(ξi = 0) = p0, P(ξi = 1) = p1, 0 < p0 < 1. Let β = 1/ρ be the golden number.
The Fibonacci expansion for a random point ρζ from [0, 1] is of the form η1ρ + η2ρ2 + … where the random variables ηk are {0, 1}-valued and ηkηk+1 = 0. The infinite random word η = η1η2 … ηn … takes values in the Fibonacci compactum and determines the so-called Erdős measure μ(A) = P(η ∈ A) on it. The invariant
Erdős measure is the shift-invariant measure with respect to which the Erdős measure is absolutely continuous.
We show that the Erdős measures are sofic. Recall that a sofic system is a symbolic system that is a continuous factor of
a topological Markov chain. A sofic measure is a one-block (or symbol-to-symbol) factor of the measure corresponding to a
homogeneous Markov chain. For the Erdős measures, the corresponding regular Markov chain has 5 states. This gives ergodic
properties of the invariant Erdős measure.
We give a new ergodic theory proof of the singularity of the distribution of the random variable ζ. Our method is also applicable
when ξ1, ξ2, … is a stationary Markov chain with values 0, 1. In particular, we prove that the distribution of ζ is singular and that
the Erdős measures appear as the result of gluing together states in a regular Markov chain with 7 states. Bibliography: 3
titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 28–47. 相似文献
3.
A. Yu. Zaįtsev 《Probability Theory and Related Fields》1988,79(2):175-200
Summary Let ξ1, ξ2,... be i.i.d random vectors in ℝ
k
with a common distribution ℒ(ξi),... = F, i = 1, 2,.... Let S
n
= ξ1+...+ξ
n
. We investigate how small is the difference between ℒ(S
n
) and ℒ(S
n+ m
) in the case when ξ
i
have symmetric distributions. 相似文献
4.
We study the structure of the distribution of a complex-valued random variable ξ = Σa
k
ξ
k
, where ξ
k
are independent complex-valued random variables with discrete distribution and a
k are terms of an absolutely convergent series. We establish a criterion of discreteness and sufficient conditions for singularity
of the distribution of ξ and investigate the fractal properties of the spectrum.
Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49,
No. 12, pp. 1653–1660, December, 1997. 相似文献
5.
László Hatvani 《Periodica Mathematica Hungarica》2008,56(1):71-82
The first part of this review paper is devoted to the simple (undamped, unforced) pendulum with a varying coefficient. If
the coefficient is a step function, then small oscillations are described by the equation
Using a probability approach, we assume that (a
k
)
k=1∞ is given, and {t
k
}
k=1∞ is chosen at random so that t
k
− t
k−1 are independent random variables. The first problem is to guarantee that all solutions tend to zero, as t → ∞, provided that a
k
↗ ∞ (k → ∞). In the problem of swinging the coefficient a
2 takes only two different values alternating each others, and t
k
− t
k−1 are identically distributed. One has to find the distributions and their critical expected values such that the amplitudes
of the oscillations tend to ∞ in some (probabilistic) sense. In the second part we deal with the damped forced pendulum equation
In 1999 J. Hubbard discovered that some motions of this simple physical model are chaotic. Recently, using also the computer
(the method of interval arithmetic), we gave a proof for Hubbard’s assertion. Here we show some tools of the proof.
Supported by the Hungarian NFSR (OTKA T49516) and by the Analysis and Stochastics Research Group of the Hungarian Academy
of Sciences. 相似文献
6.
Let ξ, ξ1, ξ2, ... be independent identically distributed random variables, and S
n
:=Σ
j=1
n
,ξ
j
, $
\bar S
$
\bar S
:= sup
n≥0
S
n
. If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $
\bar S
$
\bar S
as a → 0 and $
\bar S
$
\bar S
tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:
We obtain some results for identically and differently distributed ξ
j
. 相似文献
1. | The first model assumes that ξ j can be represented as ξ j = ζ j + αη j , where ζ1, ζ 2 , ... and η 1, η 2, ... are two independent sequences of independent random variables, identically distributed in each sequence, such that supn≥0Σ j=1 n ζ j = ∞, sup n≥0Σ j=1 n η j < ∞, and $ \bar S $ \bar S < ∞ almost surely. |
2. | In the second model we consider a triangular array scheme with parameter a and assume that the right tail distribution P(ξ j ≥ t) ∼ V (t) as t→∞ depends weakly on a, while the left tail distribution is P(ξ j < −t) = W(t/a), where V and W are regularly varying functions and $ \bar S $ \bar S < ∞ almost surely for every fixed α > 0. |
7.
Let ξ,ξ
1,ξ
2,… be positive i.i.d. random variables, S=∑
j=1∞
a(j)ξ
j
, where the coefficients a(j)≥0 are such that P(S<∞)=1. We obtain an explicit form of the asymptotics of −ln P(S<x) as x→0 for the following three cases:
The research partially supported by the RFBR grants 05-01-00810 and 06-01-00738, the Russian President’s grant NSh-8980-2006.1,
and the INTAS grant 03-51-5018. The second author also supported by the Lavrentiev SB RAS grant for young scientists. 相似文献
(i) | the sequence {a(j)} is regularly varying with exponent −β<−1, and −ln P(ξ<x)=O(x −γ+δ ) as x→0 for some δ>0, where γ=1/(β−1), |
(ii) | −ln P(ξ<x) is regularly varying with exponent −γ<0 as x→0, and a(j)=O(j −β−δ ) as j→∞ for some δ>0, where γ=1/(β−1), |
(iii) | {a(j)} decreases faster than any power of j, and P(ξ<x) is regularly varying with positive exponent as x→0. |
8.
LU Chuanrong QIU Jin & XU Jianjun School of Mathematics Statistics Zhejiang University of Finance Economics Hangzhou China Department of Mathematics Zhejiang University Hangzhou China 《中国科学A辑(英文版)》2006,49(12):1788-1799
Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn Rn, where supn E|Rn| <∞and Rn = o(n~(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc. 相似文献
9.
Jiang Chaowei Yang Xiaorong 《高校应用数学学报(英文版)》2007,22(1):87-94
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
10.
Teturo Kamae Hayato Takahashi 《Annals of the Institute of Statistical Mathematics》2006,58(3):573-593
Let be random variables as functions of β in the probability space [0,1) with the Lebesgue measure, where is considered to be an unknown parameter which we want to estimate from the observation ξ :=ξ1, ξ2...ξ
m
. Let an observation ξ be given, which is a finite Sturmian sequence. We determine the likelihood function P
α(ξ) as a function of parameter α, and obtain the maximum likelihood estimator as the relative frequency of 1s in a minimal cycle of ξ, where a factor η of ξ is called a minimal cycle if ξ is a factor
of η∞ and η has the minimum length among them. We also obtain a minimum sufficient statistics. The sample mean (ξ1 + ξ2 + ... + ξ
m
)/m which is an unbiased estimator of α is not admissible if m=6 or m ≥ 8 since it is not based on the minimum sufficient statistics. 相似文献
11.
Yu Miao 《Acta Appl Math》2009,106(2):177-184
Let X
k
=∑
i=−∞∞
a
i
ξ
k−i
,k≥1, be the moving average processes, where (ξ
i
)
i∈ℤ is a sequence of real stationary random variables. Under the assumptions that the large deviation principle (LDP) for real
stationary sequence holds, LDP for the moving average processes of real stationary sequence is established.
相似文献
12.
A. K. Aleskeviciene 《Lithuanian Mathematical Journal》2005,45(4):359-367
Let X
1, X
2,... be independent identically distributed random variables with distribution function F, S
0 = 0, S
n
= X
1 + ⋯ + X
n
, and Sˉ
n
= max1⩽k⩽n
S
k
. We obtain large-deviation theorems for S
n
and Sˉ
n
under the condition 1 − F(x) = P{X
1 ⩾ x} = e−l(x), l(x) = x
α
L(x), α ∈ (0, 1), where L(x) is a slowly varying function as x → ∞.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 447–456, October–December, 2005. 相似文献
13.
Henrik Eriksson Kimmo Eriksson Svante Linusson Johan Wästlund 《Annals of Combinatorics》2007,11(3-4):459-470
We study the length L
k
of the shortest permutation containing all patterns of length k. We establish the bounds e
−2
k
2 < L
k
≤ (2/3 + o(1))k
2. We also prove that as k → ∞, there are permutations of length (1/4 + o(1))k
2 containing almost all patterns of length k.
Received January 2, 2007 相似文献
14.
Stefan Bergman 《Annali di Matematica Pura ed Applicata》1962,57(1):295-309
Summary The author considers a schlicht pseudo-conformal mapping of a domainB in the (z
1
, z
2
)-space onto the Reinhardt circular domainC in the (ξ
1
, ξ
2
)-space by a pair of functions (see (1)§ 1). The domainB is assumed to include the bicylinder ((2)§ 2) and to omit four planes zk=ak, zk=bk, k=1, 2. Upper bounds for a sequence of the coefficients
μv
(k)
of the developments (1) are given, see p. 304. The upper bounds depend only on ak, bk, k=1, 2, and on the radii rk of the bicylinder ((2)§ 2). The bounds are obtained by using the method of the kernel function. The result can be considered as an analogue to the
inequalities of Grunsky in the theory of functions of one complex variable.
To Enrico Bompiani on his scientific Jubiles
This paper was prepared under the sponsorship of the N. S. F. 相似文献
15.
V. Bentkus 《Lithuanian Mathematical Journal》2008,48(3):237-255
Let S
n = X
1 + ⋯ + X
n be a sum of independent random variables such that 0 ⩽ X
k ⩽ 1 for all k. Write {ie237-01} and q = 1 − p. Let 0 < t < q. In our recent paper [3], we extended the inequality of Hoeffding ([6], Theorem 1) {fx237-01} to the case where X
k are unbounded positive random variables. It was assumed that the means {ie237-02} of individual summands are known. In this
addendum, we prove that the inequality still holds if only an upper bound for the mean {ie237-03} is known and that the i.i.d.
case where {ie237-04} dominates the general non-i.i.d. case. Furthermore, we provide upper bounds expressed in terms of certain
compound Poisson distributions. Such bounds can be more convenient in applications. Our inequalities reduce to the related
Hoeffding inequalities if 0 ⩽ X
k ⩽ 1. Our conditions are X
k ⩾ 0 and {ie237-05}. In particular, X
k can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus [2]. The independence of
X
k can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities
in Hoeffding [6] and Bentkus
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-25/08. 相似文献
16.
Age-dependent branching processes in random environments 总被引:4,自引:0,他引:4
We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξn) on R , and reproduce independently new particles according to a probability law p(ξn) on N. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean EξZ(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments. 相似文献
17.
Stanislav Volkov 《Journal of Theoretical Probability》2006,19(3):691-700
Vertex-reinforced random walk is a random process which visits a site with probability proportional to the weight w
k
of the number k of previous visits. We show that if w
k
∼ k
α, then there is a large time T
0 such that after T
0 the walk visits 2, 5, or ∞ sites when α < 1, = 1, or > 1, respectively. More general results are also proven.
相似文献
18.
Let {ξ
j
; j ∈ ℤ+
d
be a centered stationary Gaussian random field, where ℤ+
d
is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝd, having nonnegative integer coordinates. For each j = (j
1
, ..., jd) in ℤ+
d
, we denote |j| = j
1
... j
d
and for m, n ∈ ℤ+
d
, define S(m, n] = Σ
m<j≤n
ζ
j
, σ2(|n−m|) = ES
2
(m, n], S
n
= S(0, n] and S
0
= 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t > 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes.
Research supported by NSERC Canada grants at Carleton University, Ottawa 相似文献
19.
Wen Jiwei Yan Yunliang 《高校应用数学学报(英文版)》2006,21(1):87-95
Let X,X1,X2 be i. i. d. random variables with EX^2+δ〈∞ (for some δ〉0). Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ* (n)=supx∈zζ(x,n),ζ(x,n) =#{0≤k≤n:[Sk]=x}. A strong approximation of ζ(n) by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ*(n) are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ*(n) is proved. 相似文献
20.
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