The Gauss−Minkowski correspondence in ℝ2 states the existence of a homeomorphism between the probability measures μ on [0,2π] such that ${\int }_0^{2\pi }{\mathrm{e}}^{ix}\mathrm{d}\mu \left(x\right)=0$ ∫ 0 2 π e ix d μ ( x ) = 0 and the compact convex sets (CCS) of the plane with perimeter 1. In this article, we bring out explicit formulas relating the border of a CCS to its probability measure. As a consequence, we show that some natural operations on CCS – for example, the Minkowski sum – have natural translations in terms of probability measure operations,...

In extremal estimation theory the estimators are local or absolute extremes of functions defined on the cartesian product of the parameter by the sample space. Assuming that these functions converge uniformly, in a convenient stochastic way, to a limit function g, set estimators for the set ∇ of absolute maxima (minima) of g are obtained under the compactness assumption that ∇ is contained in a known compact U. A strongly consistent test is presented for this assumption. Moreover, when the true...

Let $(X_n,n\ge 1),({\tilde{X}}_n,n\ge 1)$ be two sequences of i.i.d. random vectors with values in ${ℝ}^k$ and $S_n=X_1+\cdots +X_n$, ${\tilde{S}}_n={\tilde{X}}_1+\cdots +{\tilde{X}}_n$, $n\ge 1$. Assuming that $E{X}_1=E{\tilde{X}}_1$, $E|X_1{|}^2<\infty$, $E|{\tilde{X}}_1{|}^{k+2}<\infty$ and the existence of a density of ${\tilde{X}}_1$ satisfying the certain conditions we prove the following inequalities: $$v(S_n,{\tilde{S}}_n)\le cmax\{v(X_1,{\tilde{X}}_1),{\zeta }_2(X_1,{\tilde{X}}_1)\},n=1,2,\cdots ,$$ where $v$ and ${\zeta }_2$ are the total variation and Zolotarev’s metrics, respectively.

The aim of this paper is to compare various criteria leading to the central limit theorem and the weak invariance principle. These criteria are the martingale-coboundary decomposition developed by Gordin in Dokl. Akad. Nauk SSSR188 (1969), the projective criterion introduced by Dedecker in Probab. Theory Related Fields110 (1998), which was subsequently improved by Dedecker and Rio in Ann. Inst. H. Poincaré Probab. Statist.36 (2000) and the condition introduced by Maxwell and Woodroofe in Ann. Probab.28...

Let $\left\{X_{ij}\right\}$, $i,j=\cdots$, be a double array of independent and identically distributed (i.i.d.) real random variables with $E{X}_{11}=\mu$, $E|X_{11}{-\mu |}^2=1$ and $E|X_{11}{|}^{4}lt;\infty$. Consider sample covariance matrices (with/without empirical centering) $𝒮=\frac{1}{n}{\sum }_{j=1}^n({𝐬}_j-\overline{𝐬}){({𝐬}_j-\overline{𝐬})}^T$ and $𝐒=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j{𝐬}_j^T$, where $\overline{𝐬}=\frac{1}{n}{\sum }_{j=1}^n{𝐬}_j$ and ${𝐬}_j={𝐓}_n^{1/2}{(X_{1j},...,X_{pj})}^T$ with ${\left({𝐓}_n^{1/2}\right)}^2={𝐓}_n$, non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of $𝒮$ and $𝐒$ are different as $n\to \infty$ with $p/n$ approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior...

In the present paper, we have established the complete convergence for weighted sums of pairwise independent random variables, from which the rate of convergence of moving average processes is deduced.

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 paper, we establish the complete convergence and complete moment convergence of weighted sums for arrays of rowwise $\varphi$-mixing random variables, and the Baum-Katz-type result for arrays of rowwise $\varphi$-mixing random variables. As an application, the Marcinkiewicz-Zygmund type strong law of large numbers for sequences of $\varphi$-mixing random variables is obtained. We extend and complement the corresponding results of X. J. Wang, S. H. Hu (2012).