We prove the central limit theorem for the integrated square error of multivariate box-spline density estimators.

The rate of moment convergence of sample sums was investigated by Chow (1988) (in case of real-valued random variables). In 2006, Rosalsky et al. introduced and investigated this concept for case random variable with Banach-valued (called complete convergence in mean of order $p$). In this paper, we give some new results of complete convergence in mean of order $p$ and its applications to strong laws of large numbers for double arrays of random variables taking values in Banach spaces.

In this article, we study the approximation of a probability measure $\mu$ on ${ℝ}^d$ by its empirical measure ${\widehat{\mu }}_N$ interpreted as a random quantization. As error criterion we consider an averaged $p$th moment Wasserstein metric. In the case where $2plt;d$, we establish fine upper and lower bounds for the error, ahigh resolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions....

Given a probability space (Ω,, P) and a closed subset X of a Banach lattice, we consider functions f: X × Ω → X and their iterates $fⁿ:X\times {\Omega }^{\mathbb{N}}\to X$ defined by f¹(x,ω) = f(x,ω₁), $f^{n+1}(x,\omega )=f(fⁿ(x,\omega ),{\omega }_{n+1})$, and obtain theorems on the convergence (a.s. and in L¹) of the sequence (fⁿ(x,·)).

In this paper the authors study the convergence properties for arrays of rowwise pairwise negatively quadrant dependent random variables. The results extend and improve the corresponding theorems of T. C. Hu, R. L. Taylor: On the strong law for arrays and for the bootstrap mean and variance, Int. J. Math. Math. Sci 20 (1997), 375–382.