Abstract:
In this paper, we construct a moment inequality for mixing dependent random variables, it is of independent interest. As applications, the consistency of the kernel density estimation is investigated. Several limit theorems are established: First, the central limit theorems for the kernel density estimator $f_{n,K}(x)$ and its distribution function are constructed. Also, the convergence rates of $\|f_{n,K}(x)-Ef_{n,K}(x)\|_{p}$ in sup-norm loss and integral $L^{p}$-norm loss are proved. Moreover, the a.s. convergence rates of the supremum of $|f_{n,K}(x)-Ef_{n,K}(x)|$ over a compact set and the whole real line are obtained. It is showed, under suitable conditions on the mixing rates, the kernel function and the bandwidths, that the optimal rates for i.i.d. random variables are also optimal for dependent ones.

Abstract:
let x1, x2,... be a strictly stationary and negatively associated sequence of random variables with mean zero and positive, finite variance, set sn = x1+ ... + xn, mn = max1 < k < n |sk|. under appropriate moment conditions, we obtain precise rates in law of the logarithm for the moment convergence of sn and mn.

Abstract:
The precise asymptotics for partial sums S_{n}=X_{1}+X_{2}+\cdots+X_{n}of strictly stationary \rho-mixing sequences are investigated, and the limit properties and the convergence rate of series \sum\limits_{n=1}^{\infty}\varphi(n)P(|S_{n}|\geq \varepsilon H(n)) as \varepsilon\searrow 0 are given, and hence the relations between functions \varphi(n) and H(n) are revealed.

Abstract:
Effects of alloying elements on microstructure and erosion resistance of Fe-C-Cr weld surfacing layer have been studied. The experimental results show that increasing C and Cr content favors improving the erosion resistance of the layer, and the excessive C and Cr result in decreasing the erosion resistance at 90 deg. erosion. That Mo, Nb or Ti improves the erosion resistance of Fe-C-Cr weld surfacing layer is mainly attributed to increasing the amount of M7C3 and forming fine NbC or TiC in austenite matrix, but the excessive Mo, Nb or Ti is unfavorable. The addition of Mo, Nb and Ti in proper combination possesses stronger effect on improving the erosion resistance and the erosion resistance (εA) of Fe-C-Cr weld surfacing layer with fine NbC, TiC and M7C3 distributing uniformly in austenite matrix obviously increases to 2.81 at 15 deg. erosion and 2.88 at 90 deg. erosion when the layer composition is 3.05C, 20.58Cr, 1.88Mo, 2.00Nb and 1.05Ti (in wt pct).