The problem of asymmetry appears in various aspects of time series modelling. Typical examples are asymmetric time series, asymmetric error distributions and asymmetric loss functions in estimating and predicting. The paper deals with asymmetric modifications of some recursive time series methods including Kalman filtering, exponential smoothing and recursive treatment of Box-Jenkins models.

We study a continuous time growth process on the $d$-dimensional hypercubic lattice ${𝒵}^d$, which admits a phenomenological interpretation as the combustion reaction $A+B\to 2A$, where $A$ represents heat particles and $B$ inert particles. This process can be described as an interacting particle system in the following way: at time 0 a simple symmetric continuous time random walk of total jump rate one begins to move from the origin of the hypercubic lattice; then, as soon as any random walk visits a site previously...

We consider a differential equation with a random rapidly varying coefficient. The random coefficient is a gaussian process with slowly decaying correlations and compete with a periodic component. In the asymptotic framework corresponding to the separation of scales present in the problem, we prove that the solution of the differential equation converges in distribution to the solution of a stochastic differential equation driven by a classical brownian motion in some cases, by a fractional brownian...

We consider a differential equation with a random rapidly varying coefficient. The random coefficient is a Gaussian process with slowly decaying correlations and compete with a periodic component. In the asymptotic framework corresponding to the separation of scales present in the problem, we prove that the solution of the differential equation converges in distribution to the solution of a stochastic differential equation driven by a classical Brownian motion in some cases, by a fractional Brownian motion...

We study the asymptotic behavior of the empirical process when the underlying data are gaussian and exhibit seasonal long-memory. We prove that the limiting process can be quite different from the limit obtained in the case of regular long-memory. However, in both cases, the limiting process is degenerated. We apply our results to von–Mises functionals and $U$-Statistics.

We study the asymptotic behavior of the empirical process when the underlying data are Gaussian and exhibit seasonal long-memory. We prove that the limiting process can be quite different from the limit obtained in the case of regular long-memory. However, in both cases, the limiting process is degenerated. We apply our results to von–Mises functionals and U-Statistics.

Let $(X_t,t\ge 0)$ be a Lévy process started at $0$, with Lévy measure $\nu$. We consider the first passage time $T_x$ of $(X_t,t\ge 0)$ to level $x\>0$, and $K_x:=X_{T_x}-𝑥$ the overshoot and $L_x:=x-X_{T_{{𝑥}^{-}}}$ the undershoot. We first prove that the Laplace transform of the random triple $(T_x,K_x,L_x)$ satisfies some kind of integral equation. Second, assuming that $\nu$ admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution as $x\to \infty$, where $\widetilde{T_x}$ denotes a suitable renormalization of $T_x$.

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν. We consider the first passage time Tx of (Xt, t ≥ 0) to level x > 0, and Kx := XTx - x the overshoot and Lx := x- XTx- the undershoot. We first prove that the Laplace transform of the random triple (Tx,Kx,Lx) satisfies some kind of integral equation. Second, assuming that ν admits exponential moments, we show that $(\widetilde{T_x},K_x,L_x)$ converges in distribution as x → ∞, where $\widetilde{T_x}$ denotes a suitable renormalization of Tx.


We prove, by means of Malliavin calculus, the convergence in $L^2$ of some properly renormalized weighted quadratic variations of bi-fractional Brownian motion (biFBM) with parameters $H$ and $K$, when $H\<1/4$ and $K\in (0,1]$.

We study the asymptotic behaviour, as n → ∞, of the Lebesgue measure of the set $x\in K:|P_E\left(x\right)|\le t$ for a random k-dimensional subspace E ⊂ ℝⁿ and an isotropic convex body K ⊂ ℝⁿ. For k growing slowly to infinity, we prove it to be close to the suitably normalised Gaussian measure in ${ℝ}^k$ of a t-dilate of the Euclidean unit ball. Some of the results hold for a wider class of probabilities on ℝⁿ.

The P.O.T. (Peaks-Over-Threshold) approach consists of using the Generalized Pareto Distribution (GPD) to approximate the distribution of excesses over a threshold. We use the probability-weighted moments to estimate the parameters of the approximating distribution. We study the asymptotic behaviour of these estimators (in particular their asymptotic bias) and also the functional bias of the GPD as an estimate of the distribution function of the excesses. We adapt penultimate approximation results...

The P.O.T. (Peaks-Over-Threshold) approach consists of using the Generalized Pareto Distribution (GPD) to approximate the distribution of excesses over a threshold. We use the probability-weighted moments to estimate the parameters of the approximating distribution. We study the asymptotic behaviour of these estimators (in particular their asymptotic bias) and also the functional bias of the GPD as an estimate of the distribution function of the excesses. We adapt penultimate approximation results...