In this paper, the filtering problem is revisited in the basic Gaussian homogeneous linear system driven by fractional Brownian motions. We exhibit a simple approximate filter which is asymptotically optimal in the sense that, when the observation time tends to infinity, the variance of the corresponding filtering error converges to the same limit as for the exact optimal filter.

We consider a diffusion process $X_t$ smoothed with (small) sampling parameter $\epsilon$. As in Berzin, León and Ortega (2001), we consider a kernel estimate ${\widehat{\alpha }}_{\epsilon }$ with window $h\left(\epsilon \right)$ of a function $\alpha$ of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the $L^p$ deviations such as$$\frac{1}{\sqrt{h}}{\left(\frac{h}{\epsilon }\right)}^{\frac{p}{2}}\left({∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p-𝔼{∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p\right).$$

We consider a diffusion process Xt smoothed with (small) sampling parameter ε. As in Berzin, León and Ortega (2001), we consider a kernel estimate ${\widehat{\alpha }}_{\epsilon }$ with window h(ε) of a function α of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the Lp deviations such as $$\frac{1}{\sqrt{h}}{\left(\frac{h}{\epsilon }\right)}^{\frac{p}{2}}\left({∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p-\text{I}\text{E}{∥{\widehat{\alpha }}_{\epsilon }-\alpha ∥}_p^p\right).$$

Linear relations, containing measurement errors in input and output data, are taken into account in this paper. Parameters of these so-called errors-in-variables (EIV) models can be estimated by minimizing the total least squares (TLS) of the input-output disturbances. Such an estimate is highly non-linear. Moreover in some realistic situations, the errors cannot be considered as independent by nature. Weakly dependent ($\alpha$- and $\varphi$-mixing) disturbances, which are not necessarily stationary nor identically...

A discrete time model of financial market is considered. In the focus of attention is the guaranteed profit of the investor which arises when the jumps of the stock price are bounded. The limit distribution of the profit as the model becomes closer to the classic model of geometrical Brownian motion is established. It is of interest that the approximating continuous time model does not assume any such profit.

We consider the asymptotic distribution of covariate values in the quantile regression basic solution under weak assumptions. A diagnostic procedure for assessing homogeneity of the conditional densities is also proposed.

The variance of the number of lattice points inside the dilated bounded set $rD$ with random position in ${ℝ}^d$ has asymptotics $\sim r^{d-1}$ if the rotational average of the squared modulus of the Fourier transform of the set is $O\left({\rho }^{-d-1}\right)$. The asymptotics follow from Wiener’s Tauberian theorem.

In Francq and Zakoïan [4], we derived stationarity conditions for ARMA$(p,q)$ models subject to Markov switching. In this paper, we show that, under appropriate moment conditions, the powers of the stationary solutions admit weak ARMA representations, which we are able to characterize in terms of $p,q$, the coefficients of the model in each regime, and the transition probabilities of the Markov chain. These representations are potentially useful for statistical applications.

In Francq and Zakoïan [4], we derived stationarity conditions for ARMA(p,q) models subject to Markov switching. In this paper, we show that, under appropriate moment conditions, the powers of the stationary solutions admit weak ARMA representations, which we are able to characterize in terms of p,q, the coefficients of the model in each regime, and the transition probabilities of the Markov chain. These representations are potentially useful for statistical applications.