We deal with real weakly stationary processes $\{X_t,t\in \mathbb{Z}\}$ with non-positive autocorrelations $\left\{r_k\right\}$, i. e. it is assumed that $r_k\le 0$ for all $k=1,2,\cdots$. We show that such processes have some special interesting properties. In particular, it is shown that each such a process can be represented as a linear process. Sufficient conditions under which the resulting process satisfies $r_k\le 0$ for all $k=1,2,\cdots$ are provided as well.

In proteomics study, Imaging Mass Spectrometry (IMS) is an emerging and very promising new technique for protein analysis from intact biological tissues. Though it has shown great potential and is very promising for rapid mapping of protein localization and the detection of sizeable differences in protein expression, challenges remain in data processing due to the difficulty of high dimensionality and the fact that the number of input variables in...

Generalised halfspace depth function is proposed. Basic properties of this depth function including the strong consistency are studied. We show, on several examples that our depth function may be considered to be more appropriate for nonsymetric distributions or for mixtures of distributions.

Let ${ℛ}_{\alpha }$ be the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by $\alpha \in ℂ$. I give an elementary proof of the necessary and sufficient condition for ${ℛ}_{\alpha }$ to be a locally finite complex measure (= complex Radon measure).

If the space $𝒬$ of quadratic forms in ${ℝ}^n$ is splitted in a direct sum ${𝒬}_1\oplus ...\oplus {𝒬}_k$ and if $X$ and $Y$ are independent random variables of ${ℝ}^n$, assume that there exist a real number $a$ such that $E\left(X\right|X+Y)=a(X+Y)$ and real distinct numbers $b_1,...,b_k$ such that $E\left(q\left(X\right)\right|X+Y)=b_{i}q(X+Y)$ for any $q$ in ${𝒬}_i.$ We prove that this happens only when $k=2$, when ${ℝ}^n$ can be structured in a Euclidean Jordan algebra and when $X$ and $Y$ have Wishart distributions corresponding to this structure.

In this paper, it has been shown that the complex matrix variate Dirichlet type I density factors into the complex matrix variate beta type I densities. Similar result has also been derived for the complex matrix variate Dirichlet type II density. Also, by using certain matrix transformations, the complex matrix variate Dirichlet distributions have been generated from the complex matrix beta distributions. Further, several results on the product of complex Wishart and complex beta matrices with...

This paper concerns generalized quadratic forms for the multivariate case. These forms are used to test linear hypotheses of parameters for the multivariate Gauss-Markoff model with singular covariance matrix. Distributions and independence of these forms are proved.