Functional Limit Theorems for Toeplitz Quadratic Functionals of Continuous time Gaussian Stationary Processes

\noindent The paper establishes weak convergence in $C[0,1]$ of normalized stochastic processes, generated by Toeplitz type quadratic functionals of a continuous time Gaussian stationary process, exhibiting long-range dependence. Both central and non-central functional limit theorems are obtained

The Trace Problem for Toeplitz Matrices and Operators and its Impact in Probability

The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szeg\"o, "Toeplitz forms and their applications". It has then been extensively studied in the literature. In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes. The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, etc. We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applications to discrete- and continuous-time stationary time series models with various types of memory structures, such as long memory, anti-persistent and short memory

Extensions of Rosenblatt's results on the asymptotic behavior of prediction error variance for deterministic stationary sequences

One of the main problem in prediction theory of discrete-time second-order stationary processes X(t) is to describe the asymptotic behavior of the best linear mean squared prediction error in predicting X(0) given X(t); −n ≤ t ≤ −1, as n goes to infinity. This behavior depends on the regularity (deterministic or non-deterministic) of the process X(t). In his seminal paper "Some purely deterministic processes" (J. of Math. and Mech., 6(6), 801-810, 1957), M. Rosenblatt has described the asymptotic behavior of the prediction error for deterministic processes in the following two cases: (a) the spectral density f of X(t) is continuous and vanishes on an interval, (b) the spectral density f has a very high order contact with zero. He showed that in the case (a) the prediction error behaves exponentially, while in the case (b), it behaves like a power as n ⟶ ∞. In this paper, using an approach different from the one applied in Rosenblatt's paper, we describe extensions of Rosenblatt's results to broader classes of spectral densities. Examples illustrate the obtained results.Published versio

Growth inhibition of cultured cancer cells by Ribes nigrum leaf extract

The present article includes data on the possible selective cytotoxic effect of extract of Ribes nigrum L. growing at high Armenian landscape. For this purpose, different non-cancer (microglial BV-2 wild type (Wt), acyl-CoA oxidase 1 (ACOX1) deficient (Acox1−/−) and cancer (human colon adenocarcinoma HT29 and human breast cancer MCF7) cell lines were applied. R. nigrum leaf ethanol extract showed a growth inhibition effect towards HT29 and MCF7 cells started from 6 h of treatment at the concentration of 0.5 mg/mL DW. The lowest concentration (0.125 mg/mL DW) of the investigated extract expressed cytotoxicity after 72 hours following cancer cell treatment. In contrast to the cancer cells, in the case of the tested non-cancer cells, cytotoxic effect was not observed at the applied concentrations. The extract sub-cytotoxic concentration, in this case, was reported to be the 1 mg/mL DW. Further investigations are needed to confirm the selective cytotoxicity and possible action mechanisms of the leaf extract of R. nigrum

Screening revealed the strong cytotoxic activity of <i>Alchemilla smirnovii</i> and <i>Hypericum alpestre</i> ethanol extracts on different cancer cell lines

Compounds of plant origin are considered promising alternative approaches in the development of medicines for the prevention and treatment of cancer. The large diversity of herbal species still requires careful exploration as a source for new anticancer compounds. The goal of the study was to screen different herbal extracts traditionally used in Armenian folk medicine for their cytotoxic effect against some cancer cell lines, and to find the prospective plant species among them. The cytotoxicity of the plant ethanol extracts was evaluated with MTT test against HeLa (human cervical carcinoma) and A549 (human lung adenocarcinoma) cells. Antioxidant properties were assessed with DPPH free radical scavenging assay. Five of the tested ten herbal extracts exhibited significant growth-inhibiting activity on HeLa cells. Moreover, Alchemilla smirnovii and Hypericum alpestre extracts also showed potent cytotoxicity on human lung adenocarcinoma cells. These two plants possessed high antiradical activity as well. Their DPPH stoichiometric values were 0.4234 and 0.14437 respectively, meaning that 1 µg of plant extract brought the reduction of DPPH equal to the respective stoichiometric values in µg. Thus, A. smirnovii and H. alpestre extracts expressed themselves as potent cytotoxic and antioxidant agents and could have promising anticancer potential. Further evaluation of their in vivo anticancer properties has much interest

Herbs Extracts in the Treatment and Prevention of Experimental Metabolic Disorders: Synergistic Hypoglycemic Activity of Ethanol Extracts of <em>Hypericum alpestre</em> and <em>Rumex obtusifolius </em>

Hyperglycemia is a metabolic condition characterized by high blood glucose level due to the disturbances of carbohydrate, lipids and protein metabolism. These disorders lead to impaired of insulin secretion and β-cell dysfunction. It is known that continuous and long-term use of synthetic anti-diabetic drugs can have side effects including hepatic impairment. Thus, the identifications of potent hypoglycemic agents from natural sources of both medicinal and edible plants with minimum side effects, is desirable. Synergetic hypoglycemic activity in experimental animals (rabbits) was shown with ethanol extracts of a mixture of the plant species Hypericum alpestre and Rumex obtusifolius. The chemical composition of these extracts was identified by gas chromatography-mass spectrometry analysis. The plant extracts mixture and natural products extracted from medicinal plant sources have potential for treatment and prevention of diabetes mellitus and metabolic disorders