In this paper, we study in frequency domain some probabilistic and statistical properties of continuous-time version of the well-known bilinear processes driven by a standard Brownian motion. This class of processes which encompasses many commonly used processes in the literature was defined as a nonlinear stochastic differential equation which has raised considerable interest in the last few years. So, the ${\mathbb {L}}_{2}$-structure of the process is studied and its covariance function is given. These structures will lead to study the strong consistency and asymptotic normality of the Whittle estimates of the unknown parameters involved in the process. Finite sample properties are also considered through Monte Carlo experiments. In end, the model is then used to model the exchanges rate of the Algerian Dinar against the US dollar.

In this paper, I propose a natural extension of time-invariant coefficients threshold GARCH (TGARCH) processes to time-varying one, in which the associated volatility switch between different regimes due to dependency of its coefficients on unobservable (latent) time homogeneous Markov chain with finite state space (MS-TGARCH). These models are showed to be capable to capture some phenomena observed for most financial time series, among others, the asymmetric patterns, leverage effects, dependency without correlation and tail heaviness. So, some theoretical probabilistic properties of such models are discussed, in particular, we establish firstly necessary and sufficient conditions ensuring the strict stationarity and ergodicity for solution process of MS-TGARCH. Secondary, we extend the standard results for the limit theory of the popular quasi-maximum likelihood estimator (QMLE) for estimating the unknown parameters involved in model and we examine thus the strong consistency of such estimates. The finite-sample properties of QMLE are illustrated by a Monte Carlo study. Our proposed model is applied to model the exchange rates of the Algerian Dinar against the single European currency (Euro).

This paper investigates and discusses the use of information divergence, through the widely used Kullback–Leibler (KL) divergence, under the multivariate (generalized) $\gamma $-order normal distribution ($\gamma $-GND). The behavior of the KL divergence, as far as its symmetricity is concerned, is studied by calculating the divergence of $\gamma $-GND over the Student’s multivariate t-distribution and vice versa. Certain special cases are also given and discussed. Furthermore, three symmetrized forms of the KL divergence, i.e., the Jeffreys distance, the geometric-KL as well as the harmonic-KL distances, are computed between two members of the $\gamma $-GND family, while the corresponding differences between those information distances are also discussed.

In this paper, we investigate a singular Moser–Trudinger inequality involving $L^{n}$ norm in the entire Euclidean space. The blow-up procedures are used for the maximizing sequence. Then we obtain the existence of extremal functions for this singular geometric inequality in whole space. In general, $W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n)$ is a continuous embedding but not compact. But in our case we can prove that $W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^n({\mathbb {R}}^n)$ is a compact embedding. Combining the compact embedding $W^{1,n}({\mathbb {R}}^n)\hookrightarrow L^q({\mathbb {R}}^n, |x|^{-s}dx)$ for all $q\ge n$ and $0<s<n$ in [18], we establish the theorems for any $0\le \alpha <1$.

In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

Let f be a formation function and G an A-group. It is said that A acts f-hypercentrally on G if A acts f-centrally on every A-composition factor of G. In this paper, groups are investigated by f-hypercentral actions. In particular, some well-known results, including a theorem of Huppert and a theorem of Hall–Higman, are generalized.