共查询到20条相似文献,搜索用时 328 毫秒
1.
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
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.
ShuXing Chen 《中国科学A辑(英文版)》2009,52(9):1829-1843
In this paper we discuss the fundamental solution of the Keldysh type operator $
L_\alpha u \triangleq \frac{{\partial ^2 u}}
{{\partial x^2 }} + y\frac{{\partial ^2 u}}
{{\partial y^2 }} + \alpha \frac{{\partial u}}
{{\partial y}}
$
L_\alpha u \triangleq \frac{{\partial ^2 u}}
{{\partial x^2 }} + y\frac{{\partial ^2 u}}
{{\partial y^2 }} + \alpha \frac{{\partial u}}
{{\partial y}}
, which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator
with $
\alpha > - \frac{1}
{2}
$
\alpha > - \frac{1}
{2}
is obtained. It is shown that the fundamental solution for such an operator generally has stronger singularity than that
for the Tricomi operator. Particularly, the fundamental solution of the Keldysh type operator with $
\alpha < \frac{1}
{2}
$
\alpha < \frac{1}
{2}
has to be defined by using the finite part of divergent integrals in the theory of distributions. 相似文献
4.
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
$
\theta (f) = \sum\limits_{n = 0}^\infty {\zeta _n D_n (f)} forf \in Q[x;\delta ],
$
\theta (f) = \sum\limits_{n = 0}^\infty {\zeta _n D_n (f)} forf \in Q[x;\delta ],
相似文献
5.
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}
):
|