We establish some exponential inequalities for positively associated random variables without the boundedness assumption. These inequalities improve the corresponding results obtained by Oliveira 2005 . By one of the inequalities, we obtain the convergence rate n−1/2 log logn 1/2 logn 2 for the case of geometrically decreasing covariances, which closes to the optimal achievable convergence rate...

Using the “basic monotonicity property” along locally admissible trajectories, we extend to very general problems certain existing results concerning the differential inequalities verified by the value function of an optimal control problem; these differential inequalities are expressed in terms of its contingent, quasitangent, and peritangent (Clarke’s) directional derivatives and in terms of ...

As general theories, currently there are concentration inequalities (of random walk) only for the cases of independence and martingale differences. In this paper, the concentration inequalities are extended to more general situations. In terms of the theory presented in the paper, the condition of independence is ∂y ∂t = constant and martingale difference’s is ∂y ∂t = 0. This paper relaxes thes...

We investigate a weakly coupled infinite system of nonlinear strong parabolic differential–functional inequalities of the following form (1) ∂tz (t, x) < f (t, x, z(t, x), ∂xz (t, x), ∂ xxz (t, x), z), i ∈ S, in an arbitrary domain D. The right-hand sides f i of these inequalites are functionals of an unknown function z and Volterra functionals only will be regarded in this paper. We give a fun...

The integral inequalities occupy privileged position in the theory of differential and integral equations. In the recent years nonlinear integral inequalities have received considerable attention because of their important applications to a variety of problems in diverse fields of nonlinear differential and integral equations. In 1919, Gronwall [4] introduced the famous Gronwall inequality in t...

Concentration inequalities have been the subject of exciting developments during the last two decades, and have been intensively studied and used as a powerful tool in various areas. These include convex geometry, functional analysis, statistical physics, mathematical statistics, pure and applied probability theory (e.g., concentration of measure phenomena in random graphs, random matrices, and...

We construct non-linear information inequalities from Matúš’ infinite series of linear information inequalities. Each single non-linear inequality is sufficiently strong to prove that the closure of the set of all entropy functions is not polyhedral for four or more random variables, a fact that was already established using the series of linear inequalities. To the best of our knowledge, they ...

Correlation inequalities are presented for functionals of a ferromagnetic Potts model with external field, using the random-cluster representation. These results extend earlier inequalities of Ganikhodjaev–Razak and Schonmann, and yield also GKS-type inequalities when the spin-space is taken as the set of qth roots of unity.

We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M. Talagrand in the recent years. Our method is based on functional inequalities of Poincar e and logarithmic Sobolev type and iteration of these inequalities. In particular, we establish with these tools sharp deviation inequalities from the mean on norms of sums of independen...