Large deviations of sums of independent random variables
We establish a Large Deviations Principle for diffusions with Lipschitz continuous oblique reflections on regular domains. The rate functional is given as the value function of a control problem and is proved to be good. The proof is based on a viscosity solution approach. The idea consists in interpreting the probabilities as the solutions to some PDEs, make the logarithmic transform, pass to the limit, and then identify the action functional as the solution of the limiting equation.
We study a continuous-time discrete population structured by a vector of ages. Individuals reproduce asexually, age and die. The death rate takes interactions into account. Adapting the approach of Fournier and Méléard, we show that in a large population limit, the microscopic process converges to the measure-valued solution of an equation that generalizes the McKendrick-Von Foerster and Gurtin-McCamy PDEs in demography. The large deviations associated with this convergence are studied. The upper-bound...
In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy Hn=∑1≤j<k≤nωjωk1{Sj=Sk} of the polymer {S1, …, Sn} equipped with random electrical charges {ω1, …, ωn}. Our approach is based on comparison of the moments between Hn and the self-intersection local time Qn=∑1≤j<k≤n1{Sj=Sk} run by the d-dimensional random walk {Sk}. As partially needed for our main objective and partially motivated by their independent interest,...
A limit theorem in the space of continuous functions for the Dirichlet polynomialwhere denote the coefficients of the Dirichlet series expansion of the function in the half-plane
We consider the standard first passage percolation model in ℤd for d≥2. We are interested in two quantities, the maximal flow τ between the lower half and the upper half of the box, and the maximal flow ϕ between the top and the bottom of the box. A standard subadditive argument yields the law of large numbers for τ in rational directions. Kesten and Zhang have proved the law of large numbers for τ and ϕ when the sides of the box are parallel to the coordinate hyperplanes: the two variables grow...
Equip the edges of the lattice ℤ2 with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in ℝ2 when the side lengths of the rectangle go to infinity. We prove that the lower large deviations are of surface order, and we prove the corresponding large deviation principle from below. This extends and improves previous large deviations results of Grimmett and Kesten [9] obtained for boxes of particular orientation.