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11.
Fix any \(n\ge 1\). Let \(\tilde{X}_1,\ldots ,\tilde{X}_n\) be independent random variables. For each \(1\le j \le n\), \(\tilde{X}_j\) is transformed in a canonical manner into a random variable \(X_j\). The \(X_j\) inherit independence from the \(\tilde{X}_j\). Let \(s_y\) and \(s_y^*\) denote the upper \(\frac{1}{y}{\underline{\text{ th }}}\) quantile of \(S_n=\sum _{j=1}^nX_j\) and \(S^*_n=\sup _{1\le k\le n}S_k\), respectively. We construct a computable quantity \(\underline{Q}_y\) based on the marginal distributions of \(X_1,\ldots ,X_n\) to produce upper and lower bounds for \(s_y\) and \(s_y^*\). We prove that for \(y\ge 8\) where and \(w_y\) is the unique solution of for \(w_y>\ln (\frac{y}{y-2})\), and for \(y\ge 37\) where The distribution of \(S_n\) is approximately centered around zero in that \(P(S_n\ge 0) \ge \frac{1}{18}\) and \(P(S_n\le 0)\ge \frac{1}{65}\). The results extend to \(n=\infty \) if and only if for some (hence all) \(a>0\)
相似文献
$$\begin{aligned} 6^{-1} \gamma _{3y/16}\underline{Q}_{3y/16}\le s^*_{y}\le \underline{Q}_y \end{aligned}$$
$$\begin{aligned} \gamma _y=\frac{1}{2w_y+1} \end{aligned}$$
$$\begin{aligned} \Big (\frac{w_y}{e\ln (\frac{y}{y-2})}\Big )^{w_y}=2y-4 \end{aligned}$$
$$\begin{aligned} \frac{1}{9}\gamma _{u(y)}\underline{Q}_{u(y)}<s_y \le \underline{Q}_y \end{aligned}$$
$$\begin{aligned} u(y)=\frac{3y}{32} \left( 1+\sqrt{1-\frac{64}{3y}}\right) . \end{aligned}$$
$$\begin{aligned} \sum _{j=1}^{\infty }E\{(\tilde{X}_j-m_j)^2\wedge a^2\}<\infty . \end{aligned}$$
(1)
12.
13.
Optics and Spectroscopy - The backscattering of light from spherical surfaces characterized by one and two-scale roughness reliefs has been investigated. The analysis is performed using the... 相似文献
14.
Fix any n≥1. Let X
1,…,X
n
be independent random variables such that S
n
=X
1+⋅⋅⋅+X
n
, and let
S*n=sup1 £ k £ nSkS^{*}_{n}=\sup_{1\le k\le n}S_{k}
. We construct upper and lower bounds for s
y
and
sy*s_{y}^{*}
, the upper
\frac1y\frac{1}{y}
th quantiles of S
n
and
S*nS^{*}_{n}
, respectively. Our approximations rely on a computable quantity Q
y
and an explicit universal constant γ
y
, the latter depending only on y, for which we prove that
${l}\displaystyle s_y\le s_y^*\le Q_y\quad\mbox{for }y>1,\\[4pt]\displaystyle \gamma_{3y/16}Q_{3y/16}-Q_1\le s_y^*\quad\mbox{for }y>\frac{32}{3},$\begin{array}{l}\displaystyle s_y\le s_y^*\le Q_y\quad\mbox{for }y>1,\\[4pt]\displaystyle \gamma_{3y/16}Q_{3y/16}-Q_1\le s_y^*\quad\mbox{for }y>\frac{32}{3},\end{array} 相似文献
15.
Sarah H. Klass Ashley E. Truxal Tahoe A. Fiala Joseph Kelly Dang Nguyen Joel A. Finbloom David E. Wemmer Alexander Pines Matthew B. Francis 《Angewandte Chemie (Weinheim an der Bergstrasse, Germany)》2019,131(29):10053-10058
The development of sensitive and chemically selective MRI contrast agents is imperative for the early detection and diagnosis of many diseases. Conventional responsive contrast agents used in 1H MRI are impaired by the high abundance of protons in the body. 129Xe hyperCEST NMR/MRI comprises a highly sensitive complement to traditional 1H MRI because of its ability to report specific chemical environments. To date, the scope of responsive 129Xe NMR contrast agents lacks breadth in the specific detection of small molecules, which are often important markers of disease. Herein, we report the synthesis and characterization of a rotaxane‐based 129Xe hyperCEST NMR contrast agent that can be turned on in response to H2O2, which is upregulated in several disease states. Added H2O2 was detected by 129Xe hyperCEST NMR spectroscopy in the low micromolar range, as well as H2O2 produced by HEK 293T cells activated with tumor necrosis factor. 相似文献
16.
Reinhold Klass 《Journal of Geometry》1983,20(1):8-27
A surface F is reflecting the bundle of straight lines through a fixed point A. The congruence of reflected straight lines is called reflection-congruence. We study its focal points. If each point of F lies in the middle of the two focal points, F is called a reflection-minimal-surface. We determine special reflection-minimal-surfaces among the minimal-surfaces, the ruled surfaces and the rotation surfaces and give a relation between reflection-minimal-surfaces and isotropic minimal-surfaces. 相似文献
17.
Let Φ be a symmetric function, nondecreasing on [0,∞) and satisfying a Δ2 growth condition, (X
1,Y
1), (X
2,Y
2),…,(X
n
,Y
n
) be arbitrary independent random vectors such that for any given i either Y
i
=X
i
or Y
i
is independent of all the other variates. The purpose of this paper is to develop an approximation of
18.
Michael J. Klass 《Journal of Combinatorial Theory, Series A》1976,20(3):273-278
Let G be a finite group which acts on a set S. We present a method of computing the entire distribution of G-orbits of S (the number of k-element G-orbits of S for all k) in terms of the number of s ε S fixed by every σ ε H for subgroups H of G, and the Möbius function μ(·, ·) defined on the subgroup lattice of G. We deduce Burnside's lemma as a consequence of our result. 相似文献
19.
Victor H. de la Peña Michael J. Klass Tze Leung Lai 《Stochastic Processes and their Applications》2009
Multivariate self-normalized processes, for which self-normalization consists of multiplying by the inverse of a positive definite matrix (instead of dividing by a positive random variable as in the scalar case), are ubiquitous in statistical applications. In this paper we make use of a technique called “pseudo-maximization” to derive exponential and moment inequalities, and bounds for boundary crossing probabilities, for these processes. In addition, Strassen-type laws of the iterated logarithm are developed for multivariate martingales, self-normalized by their quadratic or predictable variations. 相似文献
20.
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