Band 17, Heft 1

We study the normalized eigenvalue counting measure dσ of matrices of long-range percolation model. These are (2 n + 1) × (2 n + 1) random real symmetric matrices H = { H ( i, j )} i,j whose elements are independent random variables taking zero value with probability 1 – ψ (( i – j )/ b ), b ∈ ℝ + , where ψ is an even positive function ψ ( t ) ≤ 1 vanishing at infinity. It is shown that if the third moment of , is uniformly bounded, then the measure dσ = dσ n,b weakly converges in probability in the limit n, b → ∞, b = o ( n ), to the semicircle (or Wigner) distribution. The proof uses the resolvent technique combined with the cumulant expansions method. We show that the normalized trace of resolvent g n,b ( z ) converges in average and that the variance of g n,b ( z ) vanishes. In the second part of the paper, we estimate the rate of decreasing of the variance of g n,b ( z ) under further conditions on the moments of .

For a controlled stochastic differential equation with a finite horizon cost functional, a necessary condition for optimal control of degenerate diffusions with non-smooth coefficients is derived. The main idea is to show that the SDE admits a unique linearized version interpreted as its distributional derivative with respect to the initial condition. We use a technique of Bouleau–Hirsch on absolute continuity of probability measures in order to define the adjoint process on an extension of the initial probability space.

We study the behavior of the maximum of a series of independent observations subject to trends in both location and scale. The trend functions are arbitrary smooth functions and may be non-linear. It turns out that trend in scale dominates trend in location, at least for an upper tail of power type or gamma type.

In this paper, we study the one-dimensional transport process in the case of disbalance. In the hydrodynamic limit, this process approximates the diffusion process on a line. By using this property, we propose to apply transport processes instead of diffusion processes in some economical models, particularly in Black–Scholes formula. This application manages to avoid some drawbacks of diffusion processes.

In this paper, we study the regularity of the fractional derivative of the local time of some self-similar processes and we establish limit theorems for occupation time problems for a class of self-similar processes. Finally, we get the strong approximation analogues of our limit theorems. Fractional derivative and Hilbert transform of the local time of self-similar processes are limits of occupation time problems.

We study the distribution of a random variable where η k are independent random variables having the distributions: , P { η k = 1} = p 1 k ≥ 0, . We prove that the r.v. ξ has either pure discrete (atomic) or pure continuous distribution. In the case of discreteness of the r.v. ξ, we describe the set of all its atoms. For the continuously distributed r.v. ξ, we give the formula for the distribution function and prove the criterion for singularity of Cantor type.