Central Limit Theorems for Supercritical Superprocesses

In this paper, we establish a central limit theorem for a large class of general supercritical superprocesses with spatially dependent branching mechanisms satisfying a second moment condition. This central limit theorem generalizes and unifies all the central limit theorems obtained recently in Mi{\l}o\'{s} (2012, arXiv:1203:6661) and Ren, Song and Zhang (2013, to appear in Acta Appl. Math., DOI 10.1007/s10440-013-9837-0) for supercritical super Ornstein-Uhlenbeck processes. The advantage of this central limit theorem is that it allows us to characterize the limit Gaussian field. In the case of supercritical super Ornstein-Uhlenbeck processes with non-spatially dependent branching mechanisms, our central limit theorem reveals more independent structures of the limit Gaussian field

Central Limit Theorems for Super-OU Processes

In this paper we study supercritical super-OU processes with general branching mechanisms satisfying a second moment condition. We establish central limit theorems for the super-OU processes. In the small and crtical branching rate cases, our central limit theorems sharpen the corresponding results in the recent preprint of Milos in that the limit normal random variables in our central limit theorems are non-degenerate. Our central limit theorems in the large branching rate case are completely new. The main tool of the paper is the so called "backbone decomposition" of superprocesses

Strong law of large numbers for supercritical superprocesses under second moment condition

Suppose that $X=\{X_t, t\ge 0\}$ is a supercritical superprocess on a locally compact separable metric space $(E, m)$. Suppose that the spatial motion of $X$ is a Hunt process satisfying certain conditions and that the branching mechanism is of the form $$\psi(x,\lambda)=-a(x)\lambda+b(x)\lambda^2+\int_{(0,+\infty)}(e^{-\lambda y}-1+\lambda y)n(x,dy), \quad x\in E, \quad\lambda> 0,$$ where $a\in \mathcal{B}_b(E)$, $b\in \mathcal{B}_b^+(E)$ and $n$ is a kernel from $E$ to $(0,\infty)$ satisfying $$\sup_{x\in E}\int_0^\infty y^2 n(x,dy)<\infty.$$ Put $T_tf(x)=\mathbb{P}_{\delta_x}$. Let $\lambda_0>0$ be the largest eigenvalue of the generator $L$ of $T_t$, and $\phi_0$ and $\hat{\phi}_0$ be the eigenfunctions of $L$ and $\hat{L}$ (the dural of $L$) respectively associated with $\lambda_0$. Under some conditions on the spatial motion and the $\phi_0$-transformed semigroup of $T_t$, we prove that for a large class of suitable functions $f$, we have $$\lim_{t\rightarrow\infty}e^{-\lambda_0 t}< f, X_t> = W_\infty\int_E\hat{\phi}_0(y)f(y)m(dy),\quad \mathbb{P}_{\mu}{-a.s.},$$ for any finite initial measure $\mu$ on $E$ with compact support, where $W_\infty$ is the martingale limit defined by $W_\infty:=\lim_{t\to\infty}e^{-\lambda_0t}$. Moreover, the exceptional set in the above limit does not depend on the initial measure $\mu$ and the function $f$

Research on Rough Set Model Based on Golden Ratio

AbstractHow to make decision with pre-defined preference-ordered criteria also depends on the environment of the problem. Dominance rough set model is suitable for preference analysis and probabilistic rough set introduces probabilistic approaches to rough sets. In this paper, new dominance rough set rough set models are given by taking golden ratio into account. Also, we present steps to make decision using new dominance rough set models

Nonlinear Analysis of Auscultation Signals in TCM Using the Combination of Wavelet Packet Transform and Sample Entropy

Auscultation signals are nonstationary in nature. Wavelet packet transform (WPT) has currently become a very useful tool in analyzing nonstationary signals. Sample entropy (SampEn) has recently been proposed to act as a measurement for quantifying regularity and complexity of time series data. WPT and SampEn were combined in this paper to analyze auscultation signals in traditional Chinese medicine (TCM). SampEns for WPT coefficients were computed to quantify the signals from qi- and yin-deficient, as well as healthy, subjects. The complexity of the signal can be evaluated with this scheme in different time-frequency resolutions. First, the voice signals were decomposed into approximated and detailed WPT coefficients. Then, SampEn values for approximated and detailed coefficients were calculated. Finally, SampEn values with significant differences in the three kinds of samples were chosen as the feature parameters for the support vector machine to identify the three types of auscultation signals. The recognition accuracy rates were higher than 90%