共查询到20条相似文献,搜索用时 421 毫秒
1.
Approximation to the function |x| plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals |x| if α = 1. We construct a Newman Type Operator rn(x) and prove max |x|≤1|xαsgn x-rn(x)|~Cn1/4e-π1/2(1/2)αn. 相似文献
2.
Complete moment and integral convergence for sums of negatively associated random variables 总被引:2,自引:0,他引:2
For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE(max1≤k≤n|Sk|^1/q-∈bn^1/qp)^+〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 (log n) ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence. 相似文献
3.
Considering the positive d-dimensional lattice point Z
+
d
(d ≥ 2) with partial ordering ≤, let {X
k: k ∈ Z
+
d
} be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $
S_n = \sum\limits_{k \leqslant n} {X_k }
$
S_n = \sum\limits_{k \leqslant n} {X_k }
, n ∈ Z
+
d
. Let σ
i
2, i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ
2. Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $
\sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right)
$
\sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right)
. We show that when l ≥ 2 and b > −l/2, E[‖X‖2(log ‖X‖)
d−2(log log ‖X‖)
b+4] < ∞ implies $
\begin{gathered}
\mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\
= \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}}
{2}} \Gamma (b + l/2)}}
{{\Gamma (l/2)(d - 1)!}} \hfill \\
\end{gathered}
$
\begin{gathered}
\mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }}
{{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\
= \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}}
{2}} \Gamma (b + l/2)}}
{{\Gamma (l/2)(d - 1)!}} \hfill \\
\end{gathered}
, where Γ(·) is the Gamma function and $
\prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }
$
\prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/
{\vphantom {1 2}} \right.
\kern-\nulldelimiterspace} 2}} }
. 相似文献
4.
Hui Liu 《数学学报(英文版)》2012,28(5):885-900
In this paper, let Σ R2n be a symmetric compact convex hypersurface which is ( r, R )- pinched with R/r (5/3)1/2 . Then Σ carries at least two elliptic symmetric closed characteristics; moreover, Σ carries at least E [ n-1/2 ] + E [ n-1/3 ] non-hyperbolic symmetric closed characteristics. 相似文献
5.
Let {X
i
}
i=1∞ be a standardized stationary Gaussian sequence with covariance function r(n) = EX
1
X
n+1, S
n
= Σ
i=1
n
X
i
, and $\bar X_n = \tfrac{{S_n }}
{n}
$\bar X_n = \tfrac{{S_n }}
{n}
. And let N
n
be the point process formed by the exceedances of random level $(\tfrac{x}
{{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}}
{{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n
$(\tfrac{x}
{{\sqrt {2\log n} }} + \sqrt {2\log n} - \tfrac{{\log (4\pi \log n)}}
{{2\sqrt {2\log n} }})\sqrt {1 - r(n)} + \bar X_n
by X
1,X
2,…, X
n
. Under some mild conditions, N
n
and S
n
are asymptotically independent, and N
n
converges weakly to a Poisson process on (0,1]. 相似文献
6.
A. A. Mogul’skiĭ 《Siberian Advances in Mathematics》2010,20(3):191-200
Let X,X(1),X(2),... be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + ... + X(n), n = 1, 2,..., and define the Markov stopping time η
y
= inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,.... In the case $
\mathbb{E}
$
\mathbb{E}
|X|3 < ∞, the following relation was obtained in [8]: $
\mathbb{P}\left( {\eta _0 = n} \right) = \frac{1}
{{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right)
$
\mathbb{P}\left( {\eta _0 = n} \right) = \frac{1}
{{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right)
as n → ∞, where the constant R and the bounded sequence ν
n
were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence
$
H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right)
$
H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right)
for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where
the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit
$
\mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right)
$
\mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right)
for every fixed y ≥ 0 under the condition
$
\mathbb{E}
$
\mathbb{E}
X
2 < ∞ only; In [1], an explicit form of the limit $
\mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right)
$
\mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right)
was found under the same condition
$
\mathbb{E}
$
\mathbb{E}
X
2 < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that
this corrected version was formulated in [8] as a conjecture. 相似文献
7.
Stevo Stević 《Siberian Mathematical Journal》2009,50(6):1098-1105
Let $
\mathbb{B}
$
\mathbb{B}
be the unit ball in ℂ
n
and let H($
\mathbb{B}
$
\mathbb{B}
) be the space of all holomorphic functions on $
\mathbb{B}
$
\mathbb{B}
. We introduce the following integral-type operator on H($
\mathbb{B}
$
\mathbb{B}
):
$
I_\phi ^g (f)(z) = \int\limits_0^1 {\operatorname{Re} f(\phi (tz))g(tz)\frac{{dt}}
{t}} ,z \in \mathbb{B},
$
I_\phi ^g (f)(z) = \int\limits_0^1 {\operatorname{Re} f(\phi (tz))g(tz)\frac{{dt}}
{t}} ,z \in \mathbb{B},
相似文献
8.
New results about some sums s
n
(k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Y, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $
\sum\limits_{l = 0}^{\tfrac{{k - 1}}
{2}} {(_l^k )F_k - 2l^S n(k,l)}
$
\sum\limits_{l = 0}^{\tfrac{{k - 1}}
{2}} {(_l^k )F_k - 2l^S n(k,l)}
are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s
n
(k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special
inverse formulas. 相似文献
9.
Chen-Lian Chuang Tsiu-Kwen Lee Cheng-Kai Liu Yuan-Tsung Tsai 《Israel Journal of Mathematics》2010,175(1):157-178
Let R be a prime ring and δ a derivation of R. Divided powers $
D_n ^{\underline{\underline {def.}} } \tfrac{1}
{{n!}}\tfrac{{d^n }}
{{dx^n }}
$
D_n ^{\underline{\underline {def.}} } \tfrac{1}
{{n!}}\tfrac{{d^n }}
{{dx^n }}
of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that $
S^{\underline{\underline {def.}} } Q[x;\delta ]
$
S^{\underline{\underline {def.}} } Q[x;\delta ]
is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form
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