Abstract:
Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zero-mean, i.i.d. random variables X_k, are negative. We show that p_n is bounded above up to universal constant by the square root of the expected absolute value of the empirical average of {X_k}. A converse bound holds whenever P(-X_1>t) is up to constant exp(-b t) for some b>0 or when P(-X_1>t) decays super-exponentially in t. Consequently, for such random variables we have that p_n decays as n^{-1/4} if X_1 has finite second moment. In contrast, we show that for any 0 < c < 1/4 there exist integrable, zero-mean random variables for which the rate of decay of p_n is n^{-c}.

Abstract:
We obtain new upper tail probabilities of $m$-times integrated Brownian motions under the uniform norm and the $L^p$ norm. For the uniform norm, Talagrand's approach is used, while for the $L^p$ norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities (large ball probabilities) for general Gaussian random variable in Banach spaces. As applications, explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of $m$-times integrated Brownian motions are presented as well.

Abstract:
We establish upper and lower bounds for the metric entropy and bracketing entropy of the class of $d$-dimensional bounded monotonic functions under $L^p$ norms. It is interesting to see that both the metric entropy and bracketing entropy have different behaviors for $pd/(d-1)$. We apply the new bounds for bracketing entropy to establish a global rate of convergence of the MLE of a $d$-dimensional monotone density.

Abstract:
Motivated from Gaussian processes, we derive the intrinsic volumes of the infinite--dimensional Brownian motion body. The method is by discretization to a class of orthoschemes. Numerical support is offered for a conjecture of Sangwine-Yager, and another conjecture is offered on the rate of decay of intrinsic volume sequences.

Abstract:
We establish global rates of convergence of the Maximum Likelihood Estimator (MLE) of a multivariate distribution function in the case of (one type of) "interval censored" data. The main finding is that the rate of convergence of the MLE in the Hellinger metric is no worse than $n^{-1/3} (\log n)^{\gamma}$ for $\gamma = (5d - 4)/6$.

Abstract:
Let $\Omega$ be a bounded closed set in $R^d$ with non-empty interior, and let ${\cal C}_r(\Omega)$ be the class of convex functions on $\Omega$ with $L^r$-norm bounded by $1$. We obtain sharp estimates of the $\epsilon$-entropy of ${\cal C}_r(\Omega)$ under $L^p(\Omega)$ metrics, $1\le p\frac{d}{d-1}$. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions.

Abstract:
The rural dwelling house is not only an important component of housing construction in China, but also significant of promotion of the construction socialism new countryside. With the rapid development of urbanization of rural China, the level of urbanization in China will be close to 60% excepted by 2020, which undoubtedly bring to the rural building industry an unprecedented opportunity for development. The main problem which rural building materials faced is lack of the product performance certification. According to characteristic of customers and tendency of building material industry, the rural building materials should possess the harmonization, except basic quality performance. This paper analyzed the factors which influence on the environmental harmonization and established a fuzzing evaluation model, based on life cycle assessment. Through validation of environmental harmonization of ceramic, fuzzing evaluation method was effective and feasible to assess the rural building material environment harmonization.

Abstract:
Let $S_n^{(2)}$ denote the iterated partial sums. That is, $S_n^{(2)}=S_1+S_2+ ... +S_n$, where $S_i=X_1+X_2+ ... s+X_i$. Assuming $X_1, X_2,....,X_n$ are integrable, zero-mean, i.i.d. random variables, we show that the persistence probabilities $$p_n^{(2)}:=\PP(\max_{1\le i \le n}S_i^{(2)}< 0) \le c\sqrt{\frac{\EE|S_{n+1}|}{(n+1)\EE|X_1|}},$$ with $c \le 6 \sqrt{30}$ (and $c=2$ whenever $X_1$ is symmetric). The converse inequality holds whenever the non-zero $\min(-X_1,0)$ is bounded or when it has only finite third moment and in addition $X_1$ is squared integrable. Furthermore, $p_n^{(2)}\asymp n^{-1/4}$ for any non-degenerate squared integrable, i.i.d., zero-mean $X_i$. In contrast, we show that for any $0 < \gamma < 1/4$ there exist integrable, zero-mean random variables for which the rate of decay of $p_n^{(2)}$ is $n^{-\gamma}$.

Abstract:
Let $\{\xi_n\}$ be a sequence of independent and identically distributed random variables. In this paper we study the comparison for two upper tail probabilities $\mathbb{P}\{\sum_{n=1}^{\infty}a_n|\xi_n|^p\geq r\}$ and $\mathbb{P}\{\sum_{n=1}^{\infty}b_n|\xi_n|^p\geq r\}$ as $r\rightarrow\infty$ with two different real series $\{a_n\}$ and $\{b_n\}.$ The first result is for Gaussian random variables $\{\xi_n\},$ and in this case these two probabilities are equivalent after suitable scaling. The second result is for more general random variables, thus a weaker form of equivalence (namely, logarithmic level) is proved.

Abstract:
This paper presents a novel blind audio digital watermark algorithm. It makes full use of the multi-resolution of DWT and the energy compression of DCT and embeds binary image watermark in the audio by quantification process. The watermark can be extracted without the original audio signal. Experimental results demonstrate the algorithm is robust and imperceptible.