We consider an estimate of the mode $\theta$ of a multivariate probability density $f$ with support in ${ℝ}^d$ using a kernel estimate $f_n$ drawn from a sample $X_1,\cdots ,X_n$. The estimate ${\theta }_n$ is defined as any $x$ in $\{X_1,\cdots ,X_n\}$ such that $f_n\left(x\right)={max}_{i=1,\cdots ,n}{f}_n\left(X_i\right)$. It is shown that ${\theta }_n$ behaves asymptotically as any maximizer ${\widehat{\theta }}_n$ of $f_n$. More precisely, we prove that for any sequence ${\left(r_n\right)}_{n\ge 1}$ of positive real numbers such that $r_n\to \infty$ and $r_n^{d}logn/n\to 0$, one has $r_n\parallel {\theta }_n-{\widehat{\theta }}_n\parallel \to 0$ in probability. The asymptotic normality of ${\theta }_n$ follows without further work.

Let ${𝕖}_t={(e_{t1},\cdots ,e_{tp})}^{\text{'}}$ be a $p$-dimensional nonnegative strict white noise with finite second moments. Let $h_{ij}\left(x\right)$ be nondecreasing functions from $[0,\infty )$ onto $[0,\infty )$ such that $h_{ij}\left(x\right)\le x$ for $i,j=1,\cdots ,p$. Let $𝕌=\left(u_{ij}\right)$ be a $p\times p$ matrix with nonnegative elements having all its roots inside the unit circle. Define a process ${𝕏}_t={(X_{t1},\cdots ,X_{tp})}^{\text{'}}$ by $$X_{tj}=u_{j1}{h}_{1j}\left(X_{t-1,1}\right)+\cdots +u_{jp}{h}_{pj}\left(X_{t-1,p}\right)+e_{tj}$$ for $j=1,\cdots ,p$. A method for estimating $𝕌$ from a realization ${𝕏}_1,\cdots ,{𝕏}_n$ is proposed. It is proved that the estimators are strongly consistent.

We consider a random walk on a homogeneous Poisson point process with energy marks. The jump rates decay exponentially in the $\alpha$-power of the jump length and depend on the energy marks via a Boltzmann-like factor. The case $\alpha =1$ corresponds to the phonon-induced Mott variable range hopping in disordered solids in the regime of strong Anderson localization. We prove that for almost every realization of the marked process, the diffusively rescaled random walk, with an arbitrary start point,...

Let $f\left(x\right)$ be a nonnegative function with its only singularity at $x=0$, e.g. $f\left(x\right)=|x{|}^{-\alpha }$, $\alpha \>0$. We study the behavior of the Wiener process $W\left(t\right)$ in left and right hand neighborhoods of level crossings by finding necessary and sufficient conditions on $f$ for the integrals of $f\left(W\right(t\left)\right)$ to be finite or infinite.

Let $F$ be a distribution function (d.f) in the domain of attraction of an extreme value distribution $H_{\gamma }$; it is well-known that $F_u\left(x\right)$, where $F_u$ is the d.f of the excesses over $u$, converges, when $u$ tends to $s_{+}\left(F\right)$, the end-point of $F$, to $G_{\gamma }\left(\frac{x}{\sigma \left(u\right)}\right)$, where $G_{\gamma }$ is the d.f. of the Generalized Pareto Distribution. We provide conditions that ensure that there exists, for $\gamma \>-1$, a function $\Lambda$ which verifies ${lim}_{u\to s_{+}\left(F\right)}\Lambda \left(u\right)=\gamma$ and is such that $\Delta \left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\Lambda \left(u\right)}(x/\sigma \left(u\right))|$ converges to $0$ faster than $d\left(u\right)={sup}_{x\in [0,s_{+}\left(F\right)-u[}|{\overline{F}}_u\left(x\right)-{\overline{G}}_{\gamma }(x/\sigma \left(u\right))|$.

We consider a one-dimensional, transient random walk in a random i.i.d. environment. The asymptotic behaviour of such random walk depends to a large extent on a crucial parameter $\kappa gt;0$ that determines the fluctuations of the process. When $0lt;\kappa lt;2$, the averaged distributions of the hitting times of the random walk converge to a $\kappa$-stable distribution. However, it was shown recently that in this case there does not exist a quenched limiting distribution of the hitting times. That is, it is not true...

We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both...

We consider a diffusion process $X$ which is observed at times $i/n$ for $i=0,1,...,n$, each observation being subject to a measurement error. All errors are independent and centered gaussian with known variance ${\rho }_n$. There is an unknown parameter within the diffusion coefficient, to be estimated. In this first paper the case when $X$ is indeed a gaussian martingale is examined: we can prove that the LAN property holds under quite weak smoothness assumptions, with an explicit limiting Fisher information. What...

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter θ̃ of the first-order autoregressive process taking values in a real separable Banach space B (ARB(1)), if it satisfies the following relation: $X{̃}_t=\theta ̃X{̃}_{t-1}+\epsilon {̃}_t$. In this paper we study the convergence in distribution of the linear operator $I(\theta {̃}_T,\theta ̃)=(\theta {̃}_T-\theta ̃)\theta {̃}^{T-2}$ for ||θ̃|| > 1 and so we construct inequalities of Bernstein type for this operator. ...

We analyze a jump processes $Z$ with a jump measure determined by a “memory” process $S$. The state space of $(Z,S)$ is the Cartesian product of the unit circle and the real line. We prove that the stationary distribution of $(Z,S)$ is the product of the uniform probability measure and a Gaussian distribution.