We consider processes Xₜ with values in $L_p(\Omega ,ℱ,P)$ and “time” index t in a subset A of the unit cube. A natural condition of boundedness of increments is assumed. We give a full characterization of the domains A for which all such processes are a.e. continuous. We use the notion of Talagrand’s majorizing measure as well as geometrical Paszkiewicz-type characteristics of the set A. A majorizing measure is constructed.

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...

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.

In this paper we consider processes Xₜ with values in $L^p$, p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type...

Let X̂=C−Y where Y is a general one-dimensional Lévy process and C an independent subordinator. Consider the times when a new supremum of X̂ is reached by a jump of the subordinator C. We give a necessary and sufficient condition in order for such times to be discrete. When this is the case and X̂ drifts to −∞, we decompose the absolute supremum of X̂ at these times, and derive a Pollaczek–Hinchin-type formula for the distribution function of the supremum.

We show that the main result of [1] on sufficiency of existence of a majorizing measure for boundedness of a stochastic process can be naturally split in two theorems, each of independent interest. The first is that the existence of a majorizing measure is sufficient for the existence of a sequence of admissible nets (as recently introduced by Talagrand [5]), and the second that the existence of a sequence of admissible nets is sufficient for sample boundedness of a stochastic process with bounded...

$S^H=\{S_t^H,t\ge 0\}$ be a sub-fractional Brownian motion with $H\in (0,1)$. We establish the existence, the joint continuity and the Hölder regularity of the local time $L^H$ of $S^H$. We will also give Chung’s form of the law of iterated logarithm for $S^H$. This results are obtained with the decomposition of the sub-fractional Brownian motion into the sum of fractional Brownian motion plus a stochastic process with absolutely continuous trajectories. This decomposition is given by Ruiz de Chavez and Tudor [10].