The Alternative to Equilibrium Existence

This paper establishes and interprets a necessary and sucient condition for existence of (countably additive) correlated equilibrium in n-person games, assuming only that utility functions are bounded, measurable. A sequence of deviation profiles is consistent if there exists a correlated strategy that makes every profile in the sequence unprofitable with respect to the sum of utilities. An equilibrium exists if and only if every sequence of deviation profiles has a consistent subsequence. This condition fails to characterize Nash equilibrium. As a direct corollary, existence of (communication) equilibrium is characterized in games with incomplete information on type spaces large enough to include the universal one. Exact conditions for existence of approximate correlated equilibrium are also obtained, as well as a value for two-person zero-sum games.correlated equilibrium, consistency, duality, discontinuous games.

Any set of irregular points has full Hausdorff dimension and full topological entropy

We prove, for subshifts of finite type, conformal repellers, and two-dimensional horseshoes, that the set of points where both the pointwise dimension, local entropy, Lyapunov exponents, and Birkhoff averages do not exist carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute the first mathematical application of this theory

Group invariant stable processes

We study symmetric α-stable (SαS) processes defined on a separable metric space T whose finite dimensional distributions are invariant un-der a group of transformations of T. This extends the classical notion of stationarity of stochastic processes. Minimal integral representation on standard Borel space S of SαS pro-cess which is group stationary (G-stationary) in the above sense corre-sponds to a group of isometries of Lα(S, μ). We show that for 1 \u3c α \u3c 2 this group of isometries is generated by a group action and a measur-able cocycle on S. The pair of group action and cocycle induced this way is unique up to an isomorphism of group actions and cohomology equivalence relation for cocycles. We show that if a group action ad-mits a Borel selector, then cocycles are cohomologous if and only if the induce the same homomorphisms on the isotropy groups. As an application we give characterization of isotropic SαS random fields on R2 in terms of their minimal representations

Uniqueness and summability of two-dimensional Walsh series and their generalizations

This paper explores a variety of questions associated with two-dimensional Vilenkin series and their special case two-dimensional Walsh series. Some of the results are an extension of known results of the one-dimensional case to those of two-dimensions. These theorems concern the uniqueness of two-dimensional Walsh series, and a Tauberian theorem,which shows the relationship between the summability and convergence of square partial sums of two-dimensional Walsh series. The final results generalize known theorems concerning the summability of two-dimensional Vilenkin-Fourier series of unbounded type. In all cases, the sequence p = (pk) is the generating sequence for the Vilenkin group Gp (pk = 2 in the Walsh case),and the sequence (P k) is given by P0 = 1 and Pk = p0p1 … pk - 1. Fundamental real analysis and measure theory are vital to the techniques used. First, a two-dimensional quasi-measure is defined and then each two-dimensional Vilenkin series is shown to have an associated two-dimensional quasi-measure. Using this last fact and newly defined two-dimensional derivates, a two dimensional variant is found to the classical result that differentiable functions with negative derivatives are decreasing. This leads to a uniqueness result which says that if a two-dimensional Walsh series, S, satisfy both a two-dimensional CS-condition for all χ ∈ G2 and the condition lim n %rarr; ∞ S2n,2n (χ) = 0 for all but countably many χ ∈ G2, then S is the zero series. Next the dyadic square partial sums are found to be very good and finite or very bad and infinite, which leads to a Tauberian style theorem. It says that on a measurable subset of [0,1)2 a two-dimensional Walsh series, S, with bounds on certain Cesaro means for each χ in the subset, then there is a function ƒ such that lim n %rarr; ∞ S2n,2n (χ) = ƒ(χ) for almost every x in the measurable set. The final set of results deals with growth estimates for two-dimensional Vilenkin-Fourier Series of unbounded type. Under The Condition of the sequence (pk) known as δ-strongly quasi bounded,a certain maximal summability operator is bounded