in this paper, some basic results concerning strict, nonstrict inequalities, local existence theorem and differential inequalities  have been proved for an ivp of first order hybrid  random differential equations with the linear perturbation of second type. a comparison theorem is proved and  applied to prove the uniqueness of random solution for the considered perturbed random differential equ...

We derive under some regular conditions an almost sure local central limit theorem for the product of partial sums of a sequence of independent identically distributed positive random variables.

Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for Galton-Watson trees.

We consider random walks (RW) in a one-dimensional i.i.d. random environment with jumps to the nearest neighbours. For almost all environments, we prove a quenched Local Limit Theorem (LLT) for the position of the walk if the diffusivity condition is satisfied. As a corollary, we obtain the annealed version of the LLT and a new proof of the theorem of Lalley which states that the distribution o...

The bond-true self-avoiding walk is a nearest neighbour random walk on Z, for which the probability of jumping along a bond of the lattice is proportional to exp(?g number of previous jumps along that bond). We prove a limit theorem for the distribution of the local time process of this walk. A consequence of the main theorem is a limit law for n ?3=2 T n where T n is the rst hitting time of th...

Let (Ω,A, μ) be a Lebesgue space and T : Ω → Ω an ergodic measure preserving automorphism with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A simila...

Convergence of a family of Markov jump processes evolving according to coagulation-fragmentation dynamics to the corresponding determin-istic equations is proved on every nite interval of existence. After obtaining local existence by the Banach xed-point theorem, we can prove global existence using the positivity of the limit function on the local existence interval.