Assume ||·|| is a norm on ℝⁿ and ||·||⁎ its dual. Consider the closed ball $T:=B_{\left|\right|·\left|\right|}(0,r)$, r > 0. Suppose φ is an Orlicz function and ψ its conjugate. We prove that for arbitrary A,B > 0 and for each Lipschitz function f on T, $su{p}_{s,t\in T}|f\left(s\right)-f\left(t\right)|\le 6AB({\int }_0^r\psi (1/A{\epsilon }^{n-1}){\epsilon }^{n-1}d\epsilon +1/\left(n\right|B_{\left|\right|·\left|\right|}(0,1)\left|\right){\int }_T\phi (1/B||\nabla f\left(u\right)\left|\right|⁎\left)du\right)$, where |·| is the Lebesgue measure on ℝⁿ. This is a strengthening of the Sobolev inequality obtained by M. Talagrand. We use this inequality to state, for a given concave, strictly increasing function η: ℝ₊ → ℝ with η(0) = 0, a necessary and sufficient condition on φ so that each...

Many statistical applications require establishing central limit theorems for sums/integrals $S_T\left(h\right)={\int }_{t\in I_T}h\left(X_t\right)\mathrm{d}t$ or for quadratic forms $Q_T\left(h\right)={\int }_{t,s\in I_T}\widehat{b}(t-s)h(X_t,X_s)\mathrm{d}s\mathrm{d}t$, where Xt is a stationary process. A particularly important case is that of Appell polynomials h(Xt) = Pm(Xt), h(Xt,Xs) = Pm,n (Xt,Xs), since the “Appell expansion rank" determines typically the type of central limit theorem satisfied by the functionals ST(h), QT(h). We review and extend here to multidimensional indices, along lines conjectured in [F. Avram and M.S. Taqqu,...

In this paper we consider the problem of estimating the intensity of a spatial homogeneous Poisson process if a part of the observations (quadrat counts) is censored. The actual problem has occurred during a court case when one of the authors was a referee for the defense.

An example of a nonzero σ-finite Borel measure μ with everywhere dense linear manifold ${}_{\mu }$ of admissible (in the sense of invariance) translation vectors is constructed in the Hilbert space ℓ₂ such that μ and any shift ${\mu }^{\left(a\right)}$ of μ by a vector $a\in \ell ₂{\setminus }_{\mu }$ are neither equivalent nor orthogonal. This extends a result established in [7].

Geometric random sums arise in various applied problems like physics, biology, economics, risk processes, stochastic finance, queuing theory, reliability models, regenerative models, etc. Their asymptotic behaviors with convergence rates become a big subject of interest. The main purpose of this paper is to study the asymptotic behaviors of normalized geometric random sums of independent and identically distributed random variables via Gnedenko's Transfer Theorem. Moreover, using the Zolotarev probability...

We show that Billard's theorem on a.s. uniform convergence of random Fourier series with independent symmetric coefficients is not true when the coefficients are only assumed to be centered independent. We give some necessary or sufficient conditions to ensure the validity of Billard's theorem in the centered case.