On upcrossing inequalities for subadditive superstationary processes

Upcrossing Inequalities for Stationary Sequences and Applications

Let g be a function which assigns to each stationary process (Xn) ∞ n=1 and to each sample X1 . . . Xn of the process a real number g(X1, . . . , Xn), which may also depend on the distribution of (Xn). We obtain effective bounds on the probability that the sequence (g(X1, . . . , Xn)) ∞ n=1 crosses a fixed interval some number of times in terms of a quantity measuring the “average sub-additivit...

Upcrossing Inequalities for Stationary Sequences and Applications to Entropy and Complexity

An empirical statistic for a class C of stationary processes is a function g which assigns to each process (Xn) ∈ C with distribution P and to each sample X1 . . . Xn of the process a real number gP (X1, . . . , Xn). We describe a condition on g which implies that the sequence (gP (X1 . . . Xn)) ∞ n=1 obeys a (universal) upcrossing inequality, that is, that the probability that this sequence fl...

From Subadditive Inequalities of Singular Values to Triangle Inequalities of Canonical Angles

On Finding Additive, Superadditive and Subadditive Set-functions Subject to Linear Inequalities

Complexity results are obtained with regard to problems of finding solutions to set of linear inequalities which are compatible with some set-functions and a prescribed intersection graph. It is shown that the additive case is SP-complete, the superadditive case is coSP-complete, and the subadditive case is in P. 0 1 1 Finding Addit,ive, Superaddit.ive and Subadditive Set-Functions Subject to L...

Harnack inequalities for jump processes

We consider a class of pure jump Markov processes in R whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.